{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 18 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 286 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 289 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 290 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 291 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 295 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" 18 296 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 297 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 300 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 301 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 302 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 303 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 304 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 305 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 306 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 307 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 308 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 309 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 310 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 311 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 312 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 313 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 314 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 315 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 316 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 317 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 318 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 319 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 320 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 321 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 322 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 323 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 324 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 325 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 326 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 327 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 328 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 329 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 330 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 331 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 332 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 333 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 334 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 335 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 336 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 337 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 338 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 339 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 340 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 341 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 342 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 343 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 344 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 345 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 346 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 347 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 348 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 349 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 350 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 351 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 352 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 353 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 354 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 355 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 356 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 357 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 358 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 359 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 360 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" 18 361 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 362 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 363 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 364 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 365 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" 18 366 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 367 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 368 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 369 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 370 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 371 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 372 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 373 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 374 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 375 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 376 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 377 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 378 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 379 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 380 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 381 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 382 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 383 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 384 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 385 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 386 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 387 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 388 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 389 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 390 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 391 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 392 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 393 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 394 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 395 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" 18 396 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 397 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 398 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 399 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 400 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" 18 401 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 402 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 403 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 404 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 405 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 18 406 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 407 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 408 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 409 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 410 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 411 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 412 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 413 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 414 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 415 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 416 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 417 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 418 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 419 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 420 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 421 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 422 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 423 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 424 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 425 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 426 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 427 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 428 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 429 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 430 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 431 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 432 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 433 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 434 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 435 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 436 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 437 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 438 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 439 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 440 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 441 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 442 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 443 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 444 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 445 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 446 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 447 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 448 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 449 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 450 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 451 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 452 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 453 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 454 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 455 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 456 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 457 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 458 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 459 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 460 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 461 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 462 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 463 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 464 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 465 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 466 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 467 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 468 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 469 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 470 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 471 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 472 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 473 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 474 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 475 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 476 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 477 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 478 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 479 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 480 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 481 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 482 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 483 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 484 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 485 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 486 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 487 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 488 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 489 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 490 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 491 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 492 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 493 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 494 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 495 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 496 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 497 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 498 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 499 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 500 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 501 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 502 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 503 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 504 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 505 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 506 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 507 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 508 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 509 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 510 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 511 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 512 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 513 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 514 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 515 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 516 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 517 "Verdana" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 518 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 519 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 520 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 521 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 522 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 523 "Verdana" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 524 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 525 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 526 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 527 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 528 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 529 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 530 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 531 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 532 "Verdana" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 533 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 534 "Verdana" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 535 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 536 "Verdana" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 537 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 538 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 539 "Verdana" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 540 "Verdana" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 541 "Verdana" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 542 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 543 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 544 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 545 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 546 "Verdana" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 547 "Verdana" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 548 "Verdana" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 549 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 550 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 551 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 552 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 553 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 554 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 555 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 556 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 557 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 558 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 559 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 560 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 561 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 562 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 563 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 564 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 565 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 566 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 567 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 568 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 569 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 570 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 571 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 572 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 573 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 574 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 575 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 576 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 577 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 578 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 579 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 580 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 581 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 582 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 583 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 584 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 585 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 586 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 587 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 588 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 589 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 590 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 591 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 592 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 593 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 594 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 595 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 596 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 597 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 598 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 599 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 600 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 601 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 602 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 603 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 604 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 605 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 606 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 607 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 608 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 609 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 610 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 611 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 612 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 613 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 614 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 615 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 616 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 617 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 618 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" 18 619 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Times" 0 14 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 0 11 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 271 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 272 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 273 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 274 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 275 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 276 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 277 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 278 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 279 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 280 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 281 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 282 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Unit 6: Matrix Algebra" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Chapter 3 3: Matrix Computations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Section 33.2: projections" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Copyright" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 3 "" 0 "" {TEXT 320 48 "C opyright * 2001 by Addison Wesley Longman, Inc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 302 "All rights reserved. No part of this publication may be reproduced, stored in a retrieval sys tem, or transmitted, in any form or by any means, electronic, mechanic al, photocopying, recording, or otherwise, without the prior written p ermission of the publisher. Printed in the United States of America." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Initializations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "with(linalg):\nwith(plots):\nwith(plottools):\nread(` pvac.txt`):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 49 "Resolution of a Vector into Orthogonal \+ Components" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "If neigher of the vectors " }{TEXT 349 1 "B" }{TEXT -1 5 " and " }{TEXT 350 1 "A" }{TEXT -1 5 " are " }{TEXT 351 1 "0" }{TEXT -1 34 ", the zero vector, we can express " }{TEXT 352 1 "B" }{TEXT -1 41 " in terms of its vector components along " }{TEXT 353 1 "A" } {TEXT -1 22 " and perpendicular to " }{TEXT 354 1 "A" }{TEXT -1 46 ". \+ Each of these vector components are called " }{TEXT 355 11 "projectio ns" }{TEXT -1 9 ", either " }{TEXT 356 1 "B" }{XPPEDIT 357 0 "``[A]" " 6#&%!G6#%\"AG" }{TEXT -1 20 ", the projection of " }{TEXT 358 1 "B" } {TEXT -1 15 " on, or along, " }{TEXT 359 1 "A" }{TEXT -1 6 ", and " } {TEXT 360 1 "B" }{XPPEDIT 361 0 "``[`_|_`*A]" "6#&%!G6#*&%$_|gr_G\"\" \"%\"AGF(" }{TEXT -1 35 ", the projection (or component) of " }{TEXT 362 1 "B" }{TEXT -1 15 " orthogonal to " }{TEXT 363 1 "A" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "Projection o f B along A" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "The vector " }{XPPEDIT 256 0 "B[A]" "6#&%\"BG6#%\"AG" } {TEXT -1 3 " (\"" }{TEXT 364 1 "B" }{TEXT -1 4 " on " }{TEXT 365 1 "A " }{TEXT -1 22 "\"), the projection of " }{TEXT 257 1 "B" }{TEXT -1 15 " on, or along, " }{TEXT 258 1 "A" }{TEXT -1 29 ", is given by (33. 1), namely," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 374 0 "B[A]" "6#&%\"BG6#%\"AG" }{TEXT -1 3 " \+ = " }{XPPEDIT 375 0 "B*`.`*A/(A*`.`*A)" "6#**%\"BG\"\"\"%\".GF%%\"AGF% *(F'F%F&F%F'F%!\"\"" }{TEXT -1 1 " " }{TEXT 376 2 "A " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "(In words, read (33.1) as: the vector " }{XPPEDIT 366 0 "B[A]" "6#&%\"BG6#%\"AG" }{TEXT -1 3 " (\"" }{TEXT 367 1 "B" }{TEXT -1 4 " on " }{TEXT 368 1 "A" }{TEXT -1 16 "\") is given by \"" }{TEXT 369 1 "B" }{TEXT -1 5 "-dot-" } {TEXT 370 1 "A" }{TEXT -1 6 " over " }{TEXT 371 1 "A" }{TEXT -1 5 "-do t-" }{TEXT 372 1 "A" }{TEXT -1 8 ", times " }{TEXT 373 1 "A" }{TEXT -1 19 "\". There are four " }{TEXT 377 1 "A" }{TEXT -1 30 "'s on the \+ right, and just one " }{TEXT 378 1 "B" }{TEXT -1 44 ". If what is \"m ore\" is \"heavier,\" then the " }{TEXT 379 1 "A" }{TEXT -1 36 "'s sin k to the bottom, and the lone " }{TEXT 380 1 "B" }{TEXT -1 27 " floats to the top. Thus, " }{TEXT 381 1 "B" }{TEXT -1 11 " is on the " } {TEXT 382 1 "A" }{TEXT -1 25 "'s, and the formula for \"" }{TEXT 383 1 "B" }{TEXT -1 4 " on " }{TEXT 384 1 "A" }{TEXT -1 14 "\" has but one " }{TEXT 385 1 "B" }{TEXT -1 14 " and multiple " }{TEXT 386 1 "A" } {TEXT -1 83 "'s. That is how the author himself continues to remember this particular formula.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "As an example, consider the two vectors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "A := vector([2,0]);\nB := vector([1,1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "graphed, alon g with " }{TEXT 321 1 "B" }{XPPEDIT 322 0 "``[A]" "6#&%!G6#%\"AG" } {TEXT -1 16 " in Figure 33.1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 469 "p1 := arrow([0,0], A, .05,. 2,.1, color=cyan):\np2 := arrow([0,0], B, .04,.2,.1,color=black):\np3 \+ := arrow([0,0],vector([1,0]), .1,.3,.2, color=red):\np4 := arrow([0,0] ,vector([0,1]), .05,.3,.2, color=green):\np5 := textplot(\{[1,1.1,`B`] ,[2,.1,`A`], [.5,-.2,`B`], [.57,-.3,`A`], [-.7,.72,`B`], [-.5,.6,`_|_A `]\}, font=[TIMES,BOLD,12]):\np6 := plot([[1,0],[1,1],[0,1]], color=bl ack, linestyle=2):\ndisplay([p||(1..6)], scaling=constrained, xtickmar ks=[0,1,2], ytickmarks=[0,1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "From the figure, the a ngle between " }{TEXT 390 1 "A" }{TEXT -1 5 " and " }{TEXT 391 1 "B" } {TEXT -1 4 " is " }{XPPEDIT 18 0 "Pi/4" "6#*&%#PiG\"\"\"\"\"%!\"\"" } {TEXT -1 19 ", so the length of " }{TEXT 387 1 "B" }{XPPEDIT 388 0 "`` [A]" "6#&%!G6#%\"AG" }{TEXT -1 16 " is found to be " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 3 "|| " }{TEXT 389 1 "B" }{TEXT -1 4 " || " }{XPPEDIT 18 0 "cos(Pi/4)=sqrt(2)" "6#/-%$cosG6#*&% #PiG\"\"\"\"\"%!\"\"-%%sqrtG6#\"\"#" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1 /sqrt(2)=1" "6#/*&\"\"\"F%-%%sqrtG6#\"\"#!\"\"F%" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "A vector \+ of this length in the direction of " }{TEXT 392 1 "A" }{TEXT -1 7 " is 1 (" }{TEXT 393 1 "A" }{TEXT -1 6 " / || " }{TEXT 394 1 "A" }{TEXT -1 7 " ||) = " }{TEXT 395 1 "i" }{TEXT -1 18 ". Alternatively, " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 266 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 396 0 "B[A]" "6#&%\"BG6#%\"AG" }{TEXT -1 3 " = " }{XPPEDIT 397 0 "B*`.`*A/(A*`.`*A)" "6#**%\"BG\"\"\"%\".GF%%\"AGF%*(F'F%F&F%F'F% !\"\"" }{TEXT -1 1 " " }{TEXT 398 4 "A = " }{XPPEDIT 18 0 "2/4" "6#*& \"\"#\"\"\"\"\"%!\"\"" }{TEXT -1 4 " (2 " }{TEXT 399 1 "i" }{TEXT -1 4 ") = " }{TEXT 400 1 "i" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 47 "Working in Maple, we can get the angle between " } {TEXT 259 1 "A" }{TEXT -1 5 " and " }{TEXT 260 1 "B" }{TEXT -1 4 " via " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "angleAB := simplify(angle(A,B));" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "With simple right-triangle trig, the length of the red arrow " }{TEXT 402 1 "B" }{XPPEDIT 401 0 "``[A]" "6#&%!G6#%\"AG" }{TEXT -1 15 " is found \+ to be" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 3 "|| " }{TEXT 261 1 "B" }{TEXT -1 4 " || " }{XPPEDIT 18 0 "cos(theta)" "6#-%$cosG6#%&thetaG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 52 "Computed in Maple, this length of this red vector is" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "red_length := norm(B,2)*cos(angleAB);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Run this l ength out along a normalized version of the vector " }{TEXT 262 1 "A" }{TEXT -1 35 ". This vector is found in Maple by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "unitizedA : = normalize(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "The red vector is then the length \+ 1 run out along the normalized version of " }{TEXT 263 1 "A" }{TEXT -1 22 ". This is computed as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "shadow_vector := evalm(red_l ength * unitizedA);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Now, we'll use the \"formula\" " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {METAFILE 96 41 41 1 "iXmGh`B:>w=Vo;>w=Z:NqPG:MZ=^e;j;Z:^ :n_;>=E:]c:=Z:f:V[v=R;>ryipyArBfl=V;n>^;UT RcETcTX[US>r:F[:B:oi:><<:N>C:US:f:D:;B:e:qi:; fyB:^qcTTUUSaEBWTSiEB_tUUURWmraPJSVg;F v=F:NB:_;Yb;=J:FBcR:M:_k>IjgIj:jZ:N@F^@F:FB:_KjfmAF:FBf:::jXaPaTXDpql`U^:;jPF:C :[Q;>:_c<;m:Ej:>:ukcG[:JSVGwR;=:iORG[:JSFK[S;=:i?Q[:n>^CUEn;xj:V[;NZCHB]J?DJ=qm;t:=j>EN;TrUEj>A^;N:=d:GvgB:;b:MVJFk; kyyY?:N;yyyyyy=B:;B:;JBB:qQBv:>:s? " 0 "" {MPLTEXT 1 0 37 "evalm(dotprod(B,A)/dotprod(A,A) * A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The operation of projecting " }{TEXT 264 1 "B" }{TEXT -1 6 " on to " }{TEXT 265 1 "A" }{TEXT -1 82 " can also be thought of as a funct ion. In Maple, this function is constructed via" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "projBA := ( B,A) -> evalm(dotprod(B,A)/dotprod(A,A)*A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "This func tion is useful if many such projections were being computed. Otherwis e, construct the occasional projection as needed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "The order of the letters \+ matters. The projection of " }{TEXT 266 1 "B" }{TEXT -1 4 " on " } {TEXT 267 1 "A" }{TEXT -1 38 " is not the same as the projection of " }{TEXT 268 1 "A" }{TEXT -1 4 " on " }{TEXT 269 1 "B" }{TEXT -1 33 ". \+ Indeed, computing both, we see" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "B_on_A := projBA(B,A);\nA_on _B := projBA(A,B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 230 "Thus, the projection operation is not commutative. Switching the order of the vectors changes which ve ctor casts the shadow. Clearly, the lengths of the projections will n ot be the same if the vectors are not the same lengths. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "Projection of B Orthogonal to A " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "The projection, or component, of " }{TEXT 270 1 "B" }{TEXT -1 18 " perpendicular to " }{TEXT 271 1 "A" }{TEXT -1 3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 272 0 "B [`_|_`*A]" "6#&%\"BG6#*&%$_|gr_G\"\"\"%\"AGF(" }{TEXT -1 5 " = " } {XPPEDIT 273 0 "B - B[A]" "6#,&%\"BG\"\"\"&F$6#%\"AG!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "a s shown in Figure 33.1, repeated here for convenience. The green arro w, the component of " }{TEXT 274 1 "B" }{TEXT -1 27 " which is perpend icular to " }{TEXT 275 1 "A" }{TEXT -1 17 ", is the vector \"" }{TEXT 276 1 "B" }{TEXT -1 18 " perpendicular to " }{TEXT 277 1 "A" }{TEXT -1 2 ".\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 411 "p1 := arrow([0,0], A, .05,.2,.1, color=cyan):\np2 \+ := arrow([0,0], B, .04,.2,.1, color=black):\np3 := arrow([0,0],vector( [1,0]), .1,.3,.2, color=red):\np4 := arrow([0,0],vector([0,1]), .05,.3 ,.2, color=green):\np5 := textplot(\{[1,1.1,`B`],[2,.1,`A`], [.5,-.2,` B`], [.57,-.3,`A`], [-.7,.72,`B`], [-.5,.6,`_|_A`]\}, font=[TIMES,BOLD ,12]):\ndisplay([p||(1..5)], scaling=constrained, xtickmarks=[0,1,2], \+ ytickmarks=[0,1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "The hard work is done in getting t he component of " }{TEXT 278 1 "B" }{TEXT -1 1 " " }{TEXT 282 5 "along " }{TEXT -1 1 " " }{TEXT 279 1 "A" }{TEXT -1 20 ". The component of \+ " }{TEXT 280 1 "B" }{TEXT -1 1 " " }{TEXT 283 13 "perpendicular" } {TEXT -1 1 " " }{TEXT 284 2 "to" }{TEXT -1 1 " " }{TEXT 281 1 "A" } {TEXT -1 30 " is obtained by a subtraction:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "BpA := evalm(B - \+ projBA(B,A));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "The sum of the components should r estore the vector " }{TEXT 285 1 "B" }{TEXT -1 1 ":" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "evalm(B_o n_A + BpA);\nprint(B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Also, observe that the componen ts are clearly perpendicular:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dotprod(B_on_A, BpA);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 34 "Projections - A Matrix Formulation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "We next recast in \+ matrix form, our work on projections along, and perpendicular to, a ve ctor." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "Projection of B along A" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "The matrix" }}{PARA 267 "" 0 "" {TEXT -1 1 " " }{TEXT 405 1 "P" }{TEXT -1 3 " = " }{XPPEDIT 404 0 "A*A ^T/(A^T*A)" "6#*(%\"AG\"\"\")F$%\"TGF%*&)F$F'F%F$F%!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "wil l project any vector onto the vector " }{TEXT 403 1 "A" }{TEXT -1 28 " . When applied to a vector " }{TEXT 286 1 "B" }{TEXT -1 14 ", we can \+ write" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 0 "" {TEXT -1 1 " " }{TEXT 413 1 "P" }{TEXT 412 1 "B" }{TEXT -1 3 " = " }{XPPEDIT 406 0 "A*A^T/(A^T*A)" "6#*(%\"AG\"\"\")F$%\"TGF%*&)F$F'F%F$F%!\"\"" } {TEXT -1 1 " " }{TEXT 407 1 "B" }{TEXT -1 3 " = " }{XPPEDIT 408 0 "``( A^T*B)/(A^T*A)" "6#*&-%!G6#*&)%\"AG%\"TG\"\"\"%\"BGF+F+*&)F)F*F+F)F+! \"\"" }{TEXT -1 1 " " }{TEXT 409 1 "A" }{TEXT -1 3 " = " }{XPPEDIT 410 0 "B*`.`*A/(A*`.`*A)" "6#**%\"BG\"\"\"%\".GF%%\"AGF%*(F'F%F&F%F'F% !\"\"" }{TEXT -1 1 " " }{TEXT 411 1 "A" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Remenber, " }{TEXT 414 1 "A" }{TEXT -1 17 " is a vector, so " }{TEXT 415 1 "A" }{XPPEDIT 18 0 "``^T" "6#)%!G%\"TG" }{TEXT 416 1 "B" }{TEXT -1 3 " = " }{TEXT 417 1 "A" }{TEXT -1 1 " " }{TEXT 420 1 "." }{TEXT -1 1 " " }{TEXT 418 1 "B" }{TEXT -1 33 " is a scalar which commutes with " }{TEXT 419 1 "A " }{TEXT -1 41 ", thereby validating the middle fraction." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 12 "Example 3 3.1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "The matrix which projects any vector " }{TEXT 421 1 "B" }{TEXT -1 16 " onto the vector" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 271 " " 0 "" {TEXT -1 1 " " }{TEXT 422 1 "A" }{TEXT -1 3 " = " }{TEXT 423 1 "i" }{TEXT -1 5 " + 2 " }{TEXT 424 1 "j" }{TEXT -1 5 " + 3 " }{TEXT 425 1 "k" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A := vector([1,2,2]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "is " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT -1 1 " " }{TEXT 426 1 "P" } {TEXT -1 3 " = " }{XPPEDIT 427 0 "A*A^T/(A^T*A)" "6#*(%\"AG\"\"\")F$% \"TGF%*&)F$F'F%F$F%!\"\"" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "MATRIX([ [1],[2],[2]])*``(1*` `*2*` `*2)/ ``(1*` `*2*` `*2)/MATRIX([[1],[2] ,[2]])=MATRIX([[1,2,2],[2,4,4],[2,4,4]])/9" "6#/**-%'MATRIXG6#7%7#\"\" \"7#\"\"#7#F,F*-%!G6#*,F*F*%#~~GF*F,F*F2F*F,F*F*-F/6#*,F*F*F2F*F,F*F2F *F,F*!\"\"-F&6#7%7#F*7#F,7#F,F6*&-F&6#7%7%F*F,F,7%F,\"\"%FC7%F,FCFCF* \"\"*F6" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "which we obtain in Maple with" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "P := evalm( A &* transpose(A)/dotprod(A,A));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Applied to th e vector " }{TEXT 435 1 "B" }{TEXT -1 3 " = " }{TEXT 436 1 "i" }{TEXT -1 3 " + " }{TEXT 437 1 "j" }{TEXT -1 3 " + " }{TEXT 438 1 "k" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "that is, to the vector" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "B := vector([1,1,1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "we get " }}{PARA 270 "" 0 "" {TEXT 434 1 "P" }{TEXT 428 1 "B" } {TEXT -1 3 " = " }{TEXT 429 1 "B" }{XPPEDIT 430 0 "``[A]" "6#&%!G6#%\" AG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "5/9" "6#*&\"\"&\"\"\"\"\"*!\"\" " }{TEXT -1 2 " (" }{TEXT 431 1 "i" }{TEXT -1 5 " + 2 " }{TEXT 432 1 " j" }{TEXT -1 5 " + 2 " }{TEXT 433 1 "k" }{TEXT -1 2 ") " }}{PARA 0 "" 0 "" {TEXT -1 15 "that is, we get" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "B_on_A := evalm(P &* B);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Of course, " }{TEXT 444 1 "B" }{XPPEDIT 445 0 "``[A]" "6# &%!G6#%\"AG" }{TEXT -1 17 " is also given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 272 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 439 0 "B*`.`*A/(A *`.`*A" "6#**%\"BG\"\"\"%\".GF%%\"AGF%*(F'F%F&F%F'F%!\"\"" }{TEXT -1 1 " " }{TEXT 440 1 "A" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "MATRIX([[1],[ 1],[1]])*`.`*MATRIX([[1],[2],[2]])/(MATRIX([[1],[2],[2]])*`.`*MATRIX([ [1],[2],[2]]))" "6#**-%'MATRIXG6#7%7#\"\"\"7#F)7#F)F)%\".GF)-F%6#7%7#F )7#\"\"#7#F2F)*(-F%6#7%7#F)7#F27#F2F)F,F)-F%6#7%7#F)7#F27#F2F)!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "MATRIX([[1],[2],[2]])=5/9" "6#/-%'MATRI XG6#7%7#\"\"\"7#\"\"#7#F+*&\"\"&F)\"\"*!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "MATRIX([[1],[2],[2]])=5/9" "6#/-%'MATRIXG6#7%7#\"\"\"7# \"\"#7#F+*&\"\"&F)\"\"*!\"\"" }{TEXT -1 2 " (" }{TEXT 441 1 "i" } {TEXT -1 5 " + 2 " }{TEXT 442 1 "j" }{TEXT -1 5 " + 2 " }{TEXT 443 1 " k" }{TEXT -1 2 ") " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "that is, by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "evalm(dotprod(B,A)/dotprod(A,A) * A );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "Projection of B Orthogonal to A" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "The projection, or component of " }{TEXT 287 1 "B" }{TEXT -1 18 " perpendicular to " } {TEXT 288 1 "A" }{TEXT -1 12 " is given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {XPPEDIT 289 0 "B[`_|_`*A]" "6#&%\"BG6#*&% $_|gr_G\"\"\"%\"AGF(" }{TEXT -1 5 " = " }{XPPEDIT 290 0 "B - B[A]" " 6#,&%\"BG\"\"\"&F$6#%\"AG!\"\"" }{TEXT -1 3 " = " }{TEXT 291 1 "B" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 1 " " }{TEXT 446 1 "P" }{TEXT 292 1 "B" }{TEXT -1 4 " = (" } {XPPEDIT 18 0 "I-P" "6#,&%\"IG\"\"\"%\"PG!\"\"" }{TEXT -1 1 ")" } {TEXT 293 4 "B = " }{XPPEDIT 18 0 "1/9" "6#*&\"\"\"F$\"\"*!\"\"" } {TEXT -1 4 " (4 " }{TEXT 447 1 "i" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``- ``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT 448 1 "j" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT 449 1 "k" }{TEXT -1 1 ")" } }{PARA 0 "" 0 "" {TEXT -1 11 "that is, by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "BpA := evalm((1-P) & * B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "where the Maple shortcut of using " }{XPPEDIT 18 0 "1-P" "6#,&\"\"\"F$%\"PG!\"\"" }{TEXT -1 16 " for the matrix " } {XPPEDIT 18 0 "I-P" "6#,&%\"IG\"\"\"%\"PG!\"\"" }{TEXT -1 30 " is too \+ convenient not to use." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "This calculation yields the component of " }{TEXT 450 1 "B" }{TEXT -1 15 " orthogonal to " }{TEXT 451 1 "A" }{TEXT -1 6 " since" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 273 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(I-P)" "6#-%!G6#,&%\"IG\"\"\"%\"PG!\"\"" } {TEXT 452 1 "B" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "MATRIX([[8/9, -2/9, \+ -2/9], [-2/9, 5/9, -4/9], [-2/9, -4/9, 5/9]])*MATRIX([[1],[1],[1]])=1/ 9" "6#/*&-%'MATRIXG6#7%7%*&\"\")\"\"\"\"\"*!\"\",$*&\"\"#F,F-F.F.,$*&F 1F,F-F.F.7%,$*&F1F,F-F.F.*&\"\"&F,F-F.,$*&\"\"%F,F-F.F.7%,$*&F1F,F-F.F .,$*&F;F,F-F.F.*&F8F,F-F.F,-F&6#7%7#F,7#F,7#F,F,*&F,F,F-F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "MATRIX([[4],[-1],[-1]])" "6#-%'MATRIXG6#7%7#\"\" %7#,$\"\"\"!\"\"7#,$F+F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "where the matrix " }{XPPEDIT 18 0 "I-P" "6#,&%\"IG\"\"\"%\"PG!\"\"" }{TEXT -1 24 " is computed in Maple \+ as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "`I-P` := evalm(1-P);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "In general, i f " }{TEXT 453 1 "P" }{TEXT -1 34 " projects onto the vector A, then \+ " }{XPPEDIT 18 0 "I-P" "6#,&%\"IG\"\"\"%\"PG!\"\"" }{TEXT -1 43 " proj ects onto the direction orthogonal to " }{TEXT 454 1 "A" }{TEXT -1 32 ". Figure 33.2 shows the vectors " }{TEXT 455 1 "B" }{TEXT -1 2 ", " } {TEXT 456 1 "A" }{TEXT -1 2 ", " }{TEXT 457 1 "B" }{XPPEDIT 458 0 "``[ A]" "6#&%!G6#%\"AG" }{TEXT -1 6 ", and " }{TEXT 459 1 "B" }{XPPEDIT 460 0 "``[`_|_`*A]" "6#&%!G6#*&%$_|gr_G\"\"\"%\"AGF(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 674 "p1 := arrow([0,0,0],A, [1,0,0], .1,.3,.1,color=cyan):\np2 := \+ arrow([0,0,0],B, [1,0,0], .1,.3,.2, color=black):\np3 := arrow([0,0,0] ,B_on_A, [1,1,1], .2,.5,.2, color=black):\np4 := arrow([0,0,0],BpA, [0 ,0,1], .1,.4,.3, color=black):\np5 := textplot3d(\{[1,2,1.8,`A`],[1,1, .8,`B`],[4/9,2/9,-1/9,`B`], [5/9,.7,11/9,`B`]\},font=[TIMES,BOLD,12], \+ color=black):\np6 := textplot3d([4/9,.35,-.2,`_|_`],font=[TIMES,ROMAN, 6], color=black):\np7 := textplot3d(\{[4/9,.6,-.28,`A`],[5/9,.9,10/9,` A`]\}, font=[TIMES,BOLD,8], color=black):\ndisplay3d([p||(1..7)],axes= frame, scaling=constrained, labels=[x,y,`z `], labelfont=[TIMES,ITALI C,12], orientation=[-30,65], tickmarks=[[0,1],[0,1,2],[0,1,2]]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 19 "Projection Matrices" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "A matrix " }{TEXT 461 1 "P" }{TEXT -1 13 " is called a " }{TEXT 462 10 "projection" }{TEXT -1 1 " " } {TEXT 463 6 "matrix" }{TEXT -1 24 " if it is symmetric and " } {XPPEDIT 18 0 "P^2=P" "6#/*$%\"PG\"\"#F%" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(If " }{TEXT 470 1 "P" }{TEXT 464 1 "B" }{TEXT -1 22 " is the projection of " }{TEXT 465 1 "B" }{TEXT -1 6 " onto " }{TEXT 466 1 "A" }{TEXT -1 7 ", then " }{TEXT 471 1 "P" }{TEXT -1 2 " (" }{TEXT 472 1 "P" }{TEXT 467 1 "B" } {TEXT -1 41 "), the projection of the projection onto " }{TEXT 468 1 " A" }{TEXT -1 10 " is still " }{TEXT 473 1 "P" }{TEXT 469 1 "B" }{TEXT -1 5 ", so " }{XPPEDIT 18 0 "P^2=P" "6#/*$%\"PG\"\"#F%" }{TEXT -1 2 ". )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The \+ matrix P from Example 33.1, that is the matrix" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "print(P);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 37 "is symmetric, as we see in Maple with" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "equal(P, \+ transpose(P));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "and a computation shows " } {XPPEDIT 18 0 "P^2=P" "6#/*$%\"PG\"\"#F%" }{TEXT -1 10 ", that is," }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "print(P,evalm(P^2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Hence, " }{TEXT 474 1 "P" } {TEXT -1 24 " is a projection matrix." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 26 "Projection onto a Subspace" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "Next, we consider the concept of projecting a vector onto a subspace generated by the b asis vectors " }{TEXT 323 1 "A" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\" " }{TEXT -1 7 ", ..., " }{TEXT 324 1 "A" }{XPPEDIT 18 0 "``[k]" "6#&%! G6#%\"kG" }{TEXT -1 34 ". For example, the projection of " }{TEXT 483 1 "B" }{TEXT -1 30 " onto the subspace spanned by " }{TEXT 479 1 " A" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 5 " and " }{TEXT 480 1 "A" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 9 " would b e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 274 "" 0 "" {TEXT -1 1 " " }{TEXT 475 1 "B" }{XPPEDIT 477 0 "``[[A[1],A[2]]]" "6#&%!G6#7$&%\"AG6# \"\"\"&F(6#\"\"#" }{TEXT -1 3 " = " }{TEXT 476 1 "B" }{XPPEDIT 18 0 "` `[[A]]" "6#&%!G6#7#%\"AG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "the component of " }{TEXT 484 1 "B " }{TEXT -1 30 " lying in the plane formed by " }{TEXT 481 1 "A" } {XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 5 " and " }{TEXT 482 1 "A" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "By calling the \+ vectors " }{TEXT 485 1 "A" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" } {TEXT -1 7 ", ..., " }{TEXT 486 1 "A" }{XPPEDIT 18 0 "``[k]" "6#&%!G6# %\"kG" }{TEXT -1 3 " a " }{TEXT 487 5 "basis" }{TEXT -1 211 ", we are \+ stating that they span the subspace (the subspace is the collection of all possible linear combinations of the basis vectors), and that they are linearly independent. This last point is very important." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " } {TEXT 478 1 "A" }{TEXT -1 45 " be the matrix whose columns are the vec tors " }{TEXT 325 1 "A" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" } {TEXT -1 7 ", ..., " }{TEXT 326 1 "A" }{XPPEDIT 18 0 "``[k]" "6#&%!G6# %\"kG" }{TEXT -1 9 " so that " }{TEXT 488 1 "A" }{TEXT -1 4 " = [" } {TEXT 327 1 "A" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 7 ", ..., " }{TEXT 328 1 "A" }{XPPEDIT 18 0 "``[k]" "6#&%!G6#%\"kG" } {TEXT -1 2 "]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "A recipe for a matrix " }{TEXT 489 1 "P" }{TEXT -1 44 " t hat projects vectors onto this subspace is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT 490 1 "P" }{TEXT -1 3 " = " }{TEXT 491 1 "A" }{TEXT -1 2 " (" }{TEXT 493 1 "A" }{XPPEDIT 18 0 "``^T" "6#) %!G%\"TG" }{TEXT 494 1 "A" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "``^`-1`" "6 #)%!G%#-1G" }{TEXT 492 1 "A" }{XPPEDIT 18 0 "``^T" "6#)%!G%\"TG" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The matrix in (33.2), namely, in" }}{PARA 275 "" 0 "" {TEXT -1 1 " " }{TEXT 496 1 "P" }{TEXT -1 3 " = " }{XPPEDIT 495 0 "A*A ^T/(A^T*A)" "6#*(%\"AG\"\"\")F$%\"TGF%*&)F$F'F%F$F%!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "can be written as " }{TEXT 502 1 "A" }{TEXT -1 2 " (" }{TEXT 503 1 "A" } {XPPEDIT 18 0 "``^T" "6#)%!G%\"TG" }{TEXT 504 1 "A" }{TEXT -1 1 ")" } {XPPEDIT 18 0 "``^`-1`" "6#)%!G%#-1G" }{TEXT 505 1 "A" }{XPPEDIT 18 0 "``^T" "6#)%!G%\"TG" }{TEXT -1 9 " because " }{TEXT 506 1 "A" } {XPPEDIT 18 0 "``^T" "6#)%!G%\"TG" }{TEXT 507 1 "A" }{TEXT -1 46 " is \+ a scalar. This suggests (33.4), that is, " }{TEXT 497 1 "P" }{TEXT -1 3 " = " }{TEXT 498 1 "A" }{TEXT -1 2 " (" }{TEXT 500 1 "A" } {XPPEDIT 18 0 "``^T" "6#)%!G%\"TG" }{TEXT 501 1 "A" }{TEXT -1 1 ")" } {XPPEDIT 18 0 "``^`-1`" "6#)%!G%#-1G" }{TEXT 499 1 "A" }{XPPEDIT 18 0 "``^T" "6#)%!G%\"TG" }{TEXT -1 45 ", is an appropriate generalization \+ of (33.2)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 12 "Example 33.2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Form a matrix which will project vectors \+ onto the subspace spanned by the vectors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "A1 := vector([1,1,0] );\nA2 := vector([0,1,0]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "These two vectors are a ba sis for the " }{TEXT 294 2 "xy" }{TEXT -1 88 "-plane. To form a matri x which projects onto this plane, begin by forming the matrix A." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "A := augment(A1,A2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Next, form " }{TEXT 508 1 "P" }{TEXT -1 14 " by the recipe" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "P := evalm(A &* inverse(tran spose(A) &* A) &* transpose(A));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Why is this r esult not surprising?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Project the vector" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "B := vector([a,b,c]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "onto the subspace spanned by " }{TEXT 329 1 "A" } {XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 5 " and " }{TEXT 330 1 "A" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "e valm(P &* B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Again, why should we not be surpri sed by this result?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Example 3.3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 21 "To Project the vector" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "B := vector([1,2,3,4]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 40 "onto the subspace spanned by the vectors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "A1 := \+ vector([1,0,-1,1]);\nA2 := vector([0,1,1,2]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "form the matrix A given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "A := augment(A1,A2);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Next, \+ construct the projection matrix " }{TEXT 509 1 "P" }{TEXT -1 24 " acco rding to the recipe" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "P := evalm(A &* inverse(transpose(A) &* A ) &* transpose(A));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Note that " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalm(P^2); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "is the same as " }{TEXT 510 1 "P" }{TEXT -1 11 ", th at is, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "equal(P,evalm(P^2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Finally, obta in the desired projection of " }{TEXT 295 1 "B" }{TEXT -1 31 " onto th is subspace spanned by " }{XPPEDIT 296 0 "A[1]" "6#&%\"AG6#\"\"\"" } {TEXT -1 5 " and " }{XPPEDIT 297 0 "A[2]" "6#&%\"AG6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "B_onA := evalm(P &* B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "The component of " }{TEXT 333 1 "B" }{TEXT -1 48 " perpendendicular to this subspac e is given by (" }{XPPEDIT 18 0 "I-P" "6#,&%\"IG\"\"\"%\"PG!\"\"" } {TEXT -1 2 ") " }{TEXT 298 1 "B" }{TEXT -1 18 ". First, compute " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "`I-P` := evalm(1-P);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "then obtain" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "BpA := evalm(`I-P` &* B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Notice that this vector is indeed \+ perpendicular to both " }{TEXT 331 1 "A" }{XPPEDIT 18 0 "``[1]" "6#&%! G6#\"\"\"" }{TEXT -1 5 " and " }{TEXT 332 1 "A" }{XPPEDIT 18 0 "``[2] " "6#&%!G6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "dotprod(BpA, A1);\ndotprod(B pA, A2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "Final Comments" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 511 1 "A" } {TEXT -1 7 " is an " }{TEXT 311 1 "r" }{TEXT -1 1 " " }{TEXT 517 1 "x " }{TEXT -1 1 " " }{TEXT 312 1 "c" }{TEXT -1 44 " matrix whose columns are independent, then " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 " " 0 "" {TEXT 512 1 "P" }{TEXT -1 3 " = " }{TEXT 513 1 "A" }{TEXT -1 2 " (" }{TEXT 514 1 "A" }{XPPEDIT 18 0 "``^T" "6#)%!G%\"TG" }{TEXT 515 1 "A" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "``^`-1`" "6#)%!G%#-1G" }{TEXT 516 1 "A" }{XPPEDIT 18 0 "``^T" "6#)%!G%\"TG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "projects onto t he " }{TEXT 525 6 "column" }{TEXT -1 1 " " }{TEXT 526 5 "space" } {TEXT -1 4 " of " }{TEXT 518 1 "A" }{TEXT -1 24 ". (The column space \+ of " }{TEXT 519 1 "A" }{TEXT -1 100 " is the span of the columns, that is, the set of all possible linear combinations of the columns of " } {TEXT 520 1 "A" }{TEXT -1 15 ".) The matrix " }{TEXT 521 1 "A" } {XPPEDIT 18 0 "``^T" "6#)%!G%\"TG" }{TEXT 522 1 "A" }{TEXT -1 23 " wil l be an invertible " }{TEXT 313 1 "c" }{TEXT -1 1 " " }{TEXT 523 1 "x " }{TEXT -1 1 " " }{TEXT 314 1 "c" }{TEXT -1 12 " matrix and " }{TEXT 524 1 "P" }{TEXT -1 37 " can be constructed from this recipe." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "If A has \+ rank " }{XPPEDIT 18 0 "k " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 33 "Minimizing Property of Projection" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "The projection o f " }{TEXT 299 1 "B" }{TEXT -1 30 " onto the subspace spanned by " } {TEXT 553 1 "A" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 7 ", ..., " }{TEXT 554 1 "A" }{XPPEDIT 18 0 "``[k]" "6#&%!G6#%\"kG" } {TEXT -1 21 " is the component of " }{TEXT 319 1 "B" }{TEXT -1 27 " in the subspace spanned by" }}{PARA 0 "" 0 "" {TEXT 555 1 "A" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 7 ", ..., " }{TEXT 556 1 "A" } {XPPEDIT 18 0 "``[k]" "6#&%!G6#%\"kG" }{TEXT -1 85 ". The component \+ orthogonal to this subspace is the shortest vector from the tip of " } {TEXT 300 1 "B" }{TEXT -1 27 " to the subspace formed by " }}{PARA 0 " " 0 "" {TEXT 557 1 "A" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 7 ", ..., " }{TEXT 558 1 "A" }{XPPEDIT 18 0 "``[k]" "6#&%!G6#%\"kG " }{TEXT -1 134 ". Figure 33.3 (constructed below, in Maple, after t he appropriate calculations have been executed), illustrates this for \+ the vectors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "B := vector([-5,7,6]);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "and " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "A1 := vector([1,2,1]);\nA2 := vector([-1,0,2]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "The matrix " }{TEXT 559 1 "A" }{TEXT -1 4 " = [" }{TEXT 334 1 " A" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 2 ", " }{TEXT 335 1 "A" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 9 "] is the n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "A := augment(A1,A2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "and " }{TEXT 560 1 "P" }{TEXT -1 3 " = " }{TEXT 561 1 "A" }{TEXT -1 2 " (" }{TEXT 562 1 "A" }{XPPEDIT 18 0 "``^T" "6#)%!G%\"TG" }{TEXT 563 1 "A" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "``^`-1`" "6#)%!G%#-1G" }{TEXT 564 1 "A" } {XPPEDIT 18 0 "``^T" "6#)%!G%\"TG" }{TEXT -1 12 " is given by" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "P := evalm(A &* inverse(transpose(A) &* A) &* transpose(A));" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "The projection of " }{TEXT 301 1 "B" }{TEXT -1 18 " onto \+ the span of " }{TEXT 336 1 "A" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\" " }{TEXT -1 5 " and " }{TEXT 337 1 "A" }{XPPEDIT 18 0 "``[2]" "6#&%!G6 #\"\"#" }{TEXT -1 16 " is the vector P" }{TEXT 302 5 "B = B" } {XPPEDIT 18 0 "``[[A]]" "6#&%!G6#7#%\"AG" }{TEXT -1 9 " given by" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "B_on_A := evalm(P &* B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "whereas the component \+ of " }{TEXT 303 1 "B" }{TEXT -1 39 " orthogonal to the subspace spanne d by " }{TEXT 338 1 "A" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" } {TEXT -1 5 " and " }{TEXT 339 1 "A" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\" \"#" }{TEXT -1 4 " is " }{TEXT 304 1 "B" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 1 " " }{TEXT 565 1 "P" } {TEXT 305 5 "B = B" }{XPPEDIT 18 0 "``[`_|_`]" "6#&%!G6#%$_|gr_G" } {XPPEDIT 18 0 "``[[A]]" "6#&%!G6#7#%\"AG" }{TEXT -1 3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "BpA := evalm(B - B_on_A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "The vector " }{TEXT 566 1 "B" } {XPPEDIT 18 0 "``[`_|_`]" "6#&%!G6#%$_|gr_G" }{XPPEDIT 18 0 "``[[A]]" "6#&%!G6#7#%\"AG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "``(I-P)" "6#-%!G6# ,&%\"IG\"\"\"%\"PG!\"\"" }{TEXT 570 1 "B" }{TEXT -1 49 " is the shorte st vctor from the plane formed by " }{TEXT 567 1 "A" }{XPPEDIT 18 0 " ``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 5 " and " }{TEXT 568 1 "A" } {XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 16 " to the tip of " }{TEXT 569 1 "B" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 16 "To see this, let" }}{PARA 276 "" 0 "" {TEXT -1 1 " " }{TEXT 572 1 "Y" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "c[1] " "6#&%\"cG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 571 1 "A" }{XPPEDIT 18 0 " ``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "c[2]" "6#&% \"cG6#\"\"#" }{TEXT -1 1 " " }{TEXT 573 1 "A" }{XPPEDIT 18 0 "``[2]" " 6#&%!G6#\"\"#" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "``(c[1]-c[2])" "6#-%! G6#,&&%\"cG6#\"\"\"F*&F(6#\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 574 1 "i " }{TEXT -1 3 " + " }{XPPEDIT 18 0 "2*c[1]" "6#*&\"\"#\"\"\"&%\"cG6#F% F%" }{TEXT -1 1 " " }{TEXT 575 1 "j" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "``(c[1]+2*c[2])" "6#-%!G6#,&&%\"cG6#\"\"\"F**&\"\"#F*&F(6#F,F*F*" } {TEXT -1 1 " " }{TEXT 576 1 "k" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Y := evalm(c[1]*A1 + c[2]*A2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "be a general vector in the plane of " }{TEXT 577 1 "A" } {XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 5 " and " }{TEXT 578 1 "A" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 14 ". By varyi ng " }{XPPEDIT 18 0 "c[1]" "6#&%\"cG6#\"\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 12 " the vector " } {TEXT 579 1 "Y" }{TEXT -1 49 " moves in the plane. Correspondingly, t he vector" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 277 "" 0 "" {TEXT -1 1 " " }{TEXT 308 1 "R" }{TEXT -1 4 " = " }{TEXT 306 1 "B" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 1 " \+ " }{TEXT 307 4 "Y = " }{XPPEDIT 18 0 "``(-5-c[1]+c[2])" "6#-%!G6#,(\" \"&!\"\"&%\"cG6#\"\"\"F(&F*6#\"\"#F," }{TEXT -1 1 " " }{TEXT 580 1 "i " }{TEXT -1 3 " + " }{XPPEDIT 18 0 "``(7-2*c[1])" "6#-%!G6#,&\"\"(\"\" \"*&\"\"#F(&%\"cG6#F(F(!\"\"" }{TEXT -1 1 " " }{TEXT 581 1 "j" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "``(6-c[1]-2*c[2])" "6#-%!G6#,(\"\"'\"\"\"&% \"cG6#F(!\"\"*&\"\"#F(&F*6#F.F(F," }{TEXT -1 1 " " }{TEXT 582 1 "k" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "R := eva lm(B - Y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "from the tip of " }{TEXT 583 1 "Y" } {TEXT -1 15 " to the tip of " }{TEXT 584 1 "B" }{TEXT -1 34 ", varies. We then seek values of " }{XPPEDIT 18 0 "c[1]" "6#&%\"cG6#\"\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 15 " which minimize" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 278 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(c[1],c[2])" "6#-%\"fG6$&%\"cG6#\" \"\"&F'6#\"\"#" }{TEXT -1 6 " = || " }{TEXT 585 1 "R" }{TEXT -1 3 " || " }{XPPEDIT 18 0 "``^2" "6#*$%!G\"\"#" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "``(-5-c[1]+c[2])^2+``(7-2*c[1])^2+``(6-c[1]-2*c[2])^2" "6#,(*$-%!G6 #,(\"\"&!\"\"&%\"cG6#\"\"\"F*&F,6#\"\"#F.F1F.*$-F&6#,&\"\"(F.*&F1F.&F, 6#F.F.F*F1F.*$-F&6#,(\"\"'F.&F,6#F.F**&F1F.&F,6#F1F.F*F1F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "( Minimizing ||" }{TEXT 310 3 " R " }{TEXT -1 2 "||" }{XPPEDIT 18 0 "``^ 2" "6#*$%!G\"\"#" }{TEXT -1 50 " is equivalent to, but simpler than, m inimizing ||" }{TEXT 586 3 " R " }{TEXT -1 4 "||.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We form the function " } {XPPEDIT 18 0 "f(c[1],c[2])" "6#-%\"fG6$&%\"cG6#\"\"\"&F'6#\"\"#" } {TEXT -1 13 " in Maple via" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f := dotprod(R,R,orthogonal);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "The techniques of calculus are sufficient for finding the desired minimizing values of " }{XPPEDIT 18 0 "c[1]" "6#&%\"cG6#\"\" \"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" } {TEXT -1 34 ". They must satisfy the equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 279 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f[c[1 ]]=0" "6#/&%\"fG6#&%\"cG6#\"\"\"\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 280 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f[c[2]]=0" "6#/&%\"fG6#&%\"cG6#\"\"#\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 22 "that is, the equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "q1 := diff(f,c[1]) = 0;\nq2 \+ := diff(f,c[2]) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "and are therefore " }{XPPEDIT 18 0 "c[1]=2,c[2]=3" "6$/&%\"cG6#\"\"\"\"\"#/&F%6#F(\"\"$" }{TEXT -1 23 " as we find in Maple by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "q := solve(\{q1,q2\},\{c[1],c[2]\}) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "The minimizing " }{TEXT 309 1 "Y" }{TEXT -1 8 " is t hen" }}{PARA 281 "" 0 "" {TEXT -1 1 " " }{TEXT 587 1 "Y" }{TEXT -1 5 " = 2 " }{TEXT 588 1 "A" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" } {TEXT -1 5 " + 3 " }{TEXT 589 1 "A" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\" \"#" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" } {TEXT 590 1 "i" }{TEXT -1 5 " + 4 " }{TEXT 591 1 "j" }{TEXT -1 5 " + 8 " }{TEXT 592 1 "k" }{TEXT -1 3 " = " }{TEXT 594 1 "P" }{TEXT 593 1 "B " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "as we see by" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "print(B_on_A, subs(q,op(Y)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The shortest \+ vector " }{TEXT 595 1 "R" }{TEXT -1 8 " is then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 282 "" 0 "" {TEXT -1 1 " " }{TEXT 596 1 "R" } {TEXT -1 3 " = " }{TEXT 597 1 "B" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``-` `" "6#,&%!G\"\"\"F$!\"\"" }{TEXT 598 1 "Y" }{TEXT -1 3 " = " }{TEXT 599 1 "B" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\" " }{TEXT 602 1 "P" }{TEXT 600 1 "B" }{TEXT -1 3 " = " }{TEXT 601 1 "B " }{XPPEDIT 18 0 "``[`_|_`]" "6#&%!G6#%$_|gr_G" }{XPPEDIT 18 0 "``[[A] ]" "6#&%!G6#7#%\"AG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "-4" "6#,$\"\"%! \"\"" }{TEXT -1 1 " " }{TEXT 603 1 "i" }{TEXT -1 5 " + 3 " }{TEXT 604 1 "j" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" } {TEXT -1 2 "2 " }{TEXT 605 1 "k" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "which we can obtain in Ma ple by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "print(BpA, subs(q,op(R)));\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Figure 3 3.3 shows " }{TEXT 344 1 "B" }{TEXT -1 2 "; " }{TEXT 340 1 "A" } {XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 5 " and " }{TEXT 341 1 "A" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 2 "; " }{TEXT 346 1 "B" }{XPPEDIT 347 0 "``[A]" "6#&%!G6#%\"AG" }{TEXT -1 20 ", the \+ projection of " }{TEXT 348 1 "B" }{TEXT -1 24 " on the plane formed by " }{TEXT 342 1 "A" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 5 " and " }{TEXT 343 1 "A" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" } {TEXT -1 23 "; and the component of " }{TEXT 345 1 "B" }{TEXT -1 41 " \+ orthogonal to the plane of projection. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 521 "p1 := arrow([0,0,0] ,B, [4,-3,2], .2,.5,.2, color=red):\np2 := arrow([0,0,0],A1, [4,-3,2], .2,.5,.3, color=black):\np3 := arrow([0,0,0],A2, [4,-3,2], .2,.5,.3, \+ color=black):\np4 := arrow([0,0,0],B_on_A, [4,-3,2], .2,.5,.2, color=g reen):\np5 := arrow(convert(B_on_A,list), BpA, [1,1,1], .4,.6,.4,color =blue):\np6 := implicitplot3d(-4*x+3*y-2*z=-.1,x=-6..2,y=-2..5,z=-1..9 , color=yellow, style=patchnogrid):\ndisplay3d([p||(1..6)], axes=none, scaling=constrained, labels=[x,y,z], labelfont=[TIMES,BOLD,14], orien tation=[135,160]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "The vector " }{TEXT 613 1 "B" } {TEXT -1 30 " is drawn in red, the vectors " }{TEXT 606 1 "A" } {XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 5 " and " }{TEXT 607 1 "A" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 40 " are in bla ck, and the plane spanned by " }{TEXT 608 1 "A" }{XPPEDIT 18 0 "``[1] " "6#&%!G6#\"\"\"" }{TEXT -1 5 " and " }{TEXT 609 1 "A" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 16 " is in yellow. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "The projection of \+ " }{TEXT 612 1 "B" }{TEXT -1 27 " onto the plane spanned by " }{TEXT 610 1 "A" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 5 " and " }{TEXT 611 1 "A" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 9 ", namely " }{TEXT 618 1 "B" }{XPPEDIT 619 0 "``[A]" "6#&%!G6#%\"AG" } {TEXT -1 40 ", is in green, and the the component of " }{TEXT 614 1 "B " }{TEXT -1 102 " orthogonal to the plane of projection is in blue. T his blue vector is the shortest value assumed by " }{TEXT 615 1 "R" } {TEXT -1 3 " = " }{TEXT 616 1 "B" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``-` `" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 1 " " }{TEXT 617 1 "Y" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }