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}}} {EXCHG {PARA 0 "" 0 "" {TEXT 442 10 "OBJECTIVES" }{TEXT 443 1 ":" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 444 1 "\225" }{TEXT -1 53 " To distinguish between two different stru ctures in " }{TEXT 445 5 "Maple" }{TEXT -1 3 ": " }{TEXT 446 6 "vecto r" }{TEXT -1 7 " and " }{TEXT 447 10 "row matrix" }{TEXT -1 1 "." }} }{EXCHG {PARA 0 "" 0 "" {TEXT 448 1 "\225" }{TEXT -1 60 " To provide \+ alternative methods of defining and inputting " }{TEXT 463 7 "vectors " }{TEXT -1 7 " with " }{TEXT 449 5 "Maple" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 487 1 "\225" }{TEXT -1 28 " To show how \+ to convert a " }{TEXT 489 6 "vector" }{TEXT -1 8 " to a " }{TEXT 488 10 "row matrix" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 450 1 "\225" }{TEXT -1 28 " To show how to convert a " }{TEXT 455 6 "vector" }{TEXT -1 8 " to a " }{TEXT 451 13 "column matrix" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 452 1 "\225" }{TEXT -1 60 " \+ To provide alternative methods of defining and inputting " }{TEXT 464 12 "row matrices" }{TEXT -1 7 " with " }{TEXT 453 5 "Maple" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 465 1 "\225" }{TEXT -1 28 " To show how to convert a " }{TEXT 466 10 "row matrix" }{TEXT -1 8 " to a " }{TEXT 467 6 "vector" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 468 1 "\225" }{TEXT -1 60 " To provide alternative me thods of defining and inputting " }{TEXT 470 15 "column matrices" } {TEXT -1 7 " with " }{TEXT 469 5 "Maple" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 454 1 "\225" }{TEXT -1 28 " To show how to conv ert a " }{TEXT 456 13 "column matrix" }{TEXT -1 8 " to a " }{TEXT 457 6 "vector" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 458 1 " \225" }{TEXT -1 76 " To provide examples of multiplication of a row m atrix and a column matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 459 1 " \225" }{TEXT -1 76 " To provide examples of multiplication of a colum n matrix and a row matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart : with(linalg, mu ltiply) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 46 "It should be noted that although the function " } {TEXT 427 6 "vector" }{TEXT -1 16 " belongs to the " }{TEXT 426 6 "lin alg" }{TEXT -1 45 " package, it need not be specified under the " } {TEXT 431 12 "with(linalg)" }{TEXT -1 49 " command since this function is also included in " }{TEXT 429 5 "Maple" }{TEXT 428 1 "\222" } {TEXT -1 3 "s " }{TEXT 430 4 "main" }{TEXT -1 10 " library." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 295 4 "N.B." }{TEXT -1 123 " In textbooks, the names \"row matrix\" o r \"row vector\" and \"column matrix\" or \"column vector\" are used i nterchangeably. In " }{TEXT 296 5 "Maple" }{TEXT -1 5 ", a " }{TEXT 297 10 "row matrix" }{TEXT -1 9 " and a " }{TEXT 298 6 "vector" } {TEXT -1 6 " are " }{TEXT 441 9 "different" }{TEXT -1 42 " objects an d the name \"vector\" denotes in " }{TEXT 320 5 "Maple" }{TEXT -1 62 " a \"row vector\". There is no object called \"column vector\" in " } {TEXT 299 5 "Maple" }{TEXT -1 21 ". Instead, the name " }{TEXT 300 13 "column matrix" }{TEXT -1 36 " is used for the structure implied. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "For example, consider " }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT -1 1 "-" }{TEXT 319 7 "element" }{TEXT -1 41 " row and column s tructures of this kind." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 307 1 "\225" }{TEXT -1 11 " A (row) " } {TEXT 301 6 "vector" }{TEXT -1 50 " is defined and input using any of the following " }{TEXT 342 4 "four" }{TEXT -1 9 " methods:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "RV := array([a, b, c]) : R V = eval(RV) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#RVG-%'vectorG6#7% %\"aG%\"bG%\"cG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "RV := array(1..3, [a, b, c]) : RV = eval(RV) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#RVG-%'vectorG6#7%% \"aG%\"bG%\"cG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "RV := convert([a, b, c], vector) : RV = eval(RV) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#RVG-%'vectorG6 #7%%\"aG%\"bG%\"cG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "RV := vector(3, [a, b, c]) : RV = eval(RV) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#RVG-%'vectorG6#7%%\" aG%\"bG%\"cG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 353 4 "N.B." }{TEXT -1 30 " Alternatively, the comman d " }{TEXT 354 11 "RV = op(RV)" }{TEXT -1 36 " may be used to displa y the vector." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 332 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 477 4 "N.B." }{TEXT -1 21 " A pplication of the " }{TEXT 473 7 "convert" }{TEXT -1 38 " function tog ether with the form name " }{TEXT 474 4 "list" }{TEXT -1 14 " and func tion " }{TEXT 481 6 "matrix" }{TEXT -1 15 " to the above " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT -1 10 "-element " }{TEXT 475 6 "vector" } {TEXT -1 13 " returns a " }{TEXT 478 1 "(" }{XPPEDIT 18 0 "1;" "6#\" \"\"" }{TEXT 479 3 " \327 " }{XPPEDIT 18 0 "3;" "6#\"\"$" }{TEXT 480 1 ")" }{TEXT -1 2 " " }{TEXT 476 10 "row matrix" }{TEXT -1 7 ", viz. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "RM := matrix(1, 3, conv ert(RV, list)) : RM = matrix(RM) ; type(RM, matrix) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#RMG-%'matrixG6#7#7%%\"aG%\"bG%\"cG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 485 7 "Boolea n" }{TEXT -1 9 " value " }{XPPMATH 20 "6#%%trueG" }{TEXT -1 41 " re turned by the type-checking function " }{TEXT 483 4 "type" }{TEXT -1 16 " verifies that [" }{TEXT 482 2 "RM" }{TEXT -1 8 "] is a " }{TEXT 484 6 "matrix" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 258 "" 0 "" {TEXT 486 5 "* * *" }{TEXT -1 0 "" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 333 4 "N.B." }{TEXT -1 21 " Application of the " }{TEXT 303 7 "conver t" }{TEXT -1 38 " function together with the form name " }{TEXT 304 6 "matrix" }{TEXT -1 15 " to the above " }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT -1 10 "-element " }{TEXT 305 6 "vector" }{TEXT -1 13 " returns a " }{TEXT 355 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 356 3 " \+ \327 " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 357 1 ")" }{TEXT -1 2 " \+ " }{TEXT 306 13 "column matrix" }{TEXT -1 7 ", viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "CM := convert(RV, matrix) : CM = matrix (CM) ; type(CM, matrix) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#CMG- %'matrixG6#7%7#%\"aG7#%\"bG7#%\"cG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %%trueG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 460 7 "Boolean" }{TEXT -1 9 " value \+ " }{XPPMATH 20 "6#%%trueG" }{TEXT -1 41 " returned by the type-checki ng function " }{TEXT 348 4 "type" }{TEXT -1 16 " verifies that [" } {TEXT 347 2 "CM" }{TEXT -1 8 "] is a " }{TEXT 358 6 "matrix" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 " " 0 "" {TEXT 334 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 308 1 "\225" }{TEXT -1 5 " A " }{TEXT 359 1 "(" }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 360 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 361 1 ")" }{TEXT -1 2 " " }{TEXT 302 10 "row matrix" }{TEXT -1 50 " is defined and input using any of \+ the following " }{TEXT 343 4 "four" }{TEXT -1 9 " methods:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "RM := array([ [a, b, c] ]) : RM = matrix(RM) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#RMG-%'matrixG6#7#7 %%\"aG%\"bG%\"cG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "RM := convert([ [a, b, c] ], matrix ) : RM = matrix(RM) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#RMG-%'ma trixG6#7#7%%\"aG%\"bG%\"cG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "RM := matrix(1, 3, [a, b, c ]) : RM = matrix(RM) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#RMG-%'m atrixG6#7#7%%\"aG%\"bG%\"cG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "RM := matrix([ [a, b, c] ] ) : RM = matrix(RM) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#RMG-%'ma trixG6#7#7%%\"aG%\"bG%\"cG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 258 "" 0 "" {TEXT 335 5 "* * *" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 336 4 "N.B." }{TEXT -1 28 " Triple application of the " }{TEXT 309 7 "convert" }{TEXT -1 39 " function together with the form names " }{TEXT 471 3 "set" } {TEXT -1 2 ", " }{TEXT 472 4 "list" }{TEXT -1 6 ", and " }{TEXT 310 6 "vector" }{TEXT -1 15 " to the above " }{TEXT 311 10 "row matrix" } {TEXT -1 19 " returns a (row) " }{TEXT 312 6 "vector" }{TEXT -1 7 ", e.g." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "RV := convert(con vert(convert(RM, set), list), vector) : RV = eval(RV) ; type(RV, ve ctor) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#RVG-%'vectorG6#7%%\"aG% \"bG%\"cG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " } {TEXT 462 7 "Boolean" }{TEXT -1 9 " value " }{XPPMATH 20 "6#%%trueG " }{TEXT -1 41 " returned by the type-checking function " }{TEXT 349 4 "type" }{TEXT -1 16 " verifies that [" }{TEXT 350 2 "RV" }{TEXT -1 14 "] is a (row) " }{TEXT 362 6 "vector" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 337 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 314 1 "\225" }{TEXT -1 5 " A " }{TEXT 363 1 "(" } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 364 3 " \327 " }{XPPEDIT 18 0 "1" " 6#\"\"\"" }{TEXT 365 1 ")" }{TEXT -1 2 " " }{TEXT 313 13 "column matr ix" }{TEXT -1 50 " is defined and input using any of the following " }{TEXT 344 4 "four" }{TEXT -1 9 " methods:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 52 "CM := array([ [a], [b], [c] ]) : CM = matrix(CM) \+ ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#CMG-%'matrixG6#7%7#%\"aG7#%\"b G7#%\"cG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "CM := convert([ [a], [b], [c] ], matrix) : \+ CM = matrix(CM) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#CMG-%'matrixG6 #7%7#%\"aG7#%\"bG7#%\"cG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "CM := matrix(3, 1, [a, b, c] ) : CM = matrix(CM) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#CMG-%'ma trixG6#7%7#%\"aG7#%\"bG7#%\"cG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "CM := matrix([ [a], [b] , [c] ]) : CM = matrix(CM) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#C MG-%'matrixG6#7%7#%\"aG7#%\"bG7#%\"cG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 338 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 339 4 "N.B." }{TEXT -1 21 " Application of the " }{TEXT 315 7 "conver t" }{TEXT -1 38 " function together with the form name " }{TEXT 316 6 "vector" }{TEXT -1 15 " to the above " }{TEXT 317 13 "column matrix" }{TEXT -1 19 " returns a (row) " }{TEXT 318 6 "vector" }{TEXT -1 7 " , e.g." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "RV := convert(CM , vector) : RV = eval(RV) ; type(RV, vector) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%#RVG-%'vectorG6#7%%\"aG%\"bG%\"cG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 461 7 "Boolean" } {TEXT -1 9 " value " }{XPPMATH 20 "6#%%trueG" }{TEXT -1 41 " return ed by the type-checking function " }{TEXT 351 4 "type" }{TEXT -1 16 " \+ verifies that [" }{TEXT 352 2 "RV" }{TEXT -1 14 "] is a (row) " } {TEXT 366 6 "vector" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 340 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 490 4 "N .B." }{TEXT -1 19 " Conversion of a " }{TEXT 491 10 "row matrix" } {TEXT -1 8 " to a " }{TEXT 492 13 "column matrix" }{TEXT -1 40 " an d vice versa is simplest using the " }{TEXT 493 13 "transposition" } {TEXT -1 12 " operation." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "[ For the " }{TEXT 495 9 "transpo se" }{TEXT -1 36 " of a matrix, refer to Unit (10). ]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 494 5 "* * *" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 259 "" 0 "" {TEXT 293 1 "A" }{TEXT -1 2 ". " }{TEXT 269 50 "Multiplication of a row matrix and a column matrix" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "A ccording to the multiplication conformability rule, the product of a \+ " }{TEXT 367 1 "(" }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 368 3 " \327 \+ " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 369 1 ")" }{TEXT -1 2 " " } {TEXT 258 10 "row matrix" }{TEXT -1 3 " [" }{TEXT 263 2 "RM" }{TEXT -1 10 "] and an " }{TEXT 370 1 "(" }{XPPEDIT 18 0 "m" "6#%\"mG" } {TEXT 371 3 " \327 " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 372 1 ")" } {TEXT -1 2 " " }{TEXT 259 13 "column matrix" }{TEXT -1 3 " [" } {TEXT 264 2 "CM" }{TEXT -1 122 "] is possible only, if the number of c olumns in the row matrix is equal to the number of rows in the column \+ matrix, i.e. " }{XPPEDIT 18 0 "n = m" "6#/%\"nG%\"mG" }{TEXT -1 20 ". The result is a " }{TEXT 278 6 "scalar" }{TEXT -1 34 " (number), w hich is displayed in " }{TEXT 279 5 "Maple" }{TEXT -1 7 " as a " } {TEXT 373 1 "(" }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 374 3 " \327 " } {XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 375 1 ")" }{TEXT -1 61 " matrix. \+ This special form of product is called either the " }{TEXT 280 5 "inn er" }{TEXT -1 18 " product or the " }{TEXT 281 6 "scalar" }{TEXT -1 16 " product of a " }{TEXT 291 10 "row matrix" }{TEXT -1 9 " and a \+ " }{TEXT 292 13 "column matrix" }{TEXT -1 87 ". [ Refer also to Unit (4) for the concept of inner product used in a similar sense. ]" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " }{TEXT 376 1 "(" }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 377 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 378 1 ")" }{TEXT -1 14 " row matrix [" }{TEXT 267 2 "RM" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "RM := mat rix(1, 3, [rm[11], rm[12], rm[13]]) : RM = matrix(RM) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#RMG-%'matrixG6#7#7%&%#rmG6#\"#6&F+6#\"#7&F+6 #\"#8" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "and a " }{TEXT 379 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 380 3 " \327 " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 381 1 ")" } {TEXT -1 17 " column matrix [" }{TEXT 268 2 "CM" }{TEXT -1 10 "] give n as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "CM := matrix(3, 1, \+ [cm[11], cm[21], cm[31]]) : CM = matrix(CM) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#CMG-%'matrixG6#7%7#&%#cmG6#\"#67#&F+6#\"#@7#&F+6#\"# J" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The product [" }{TEXT 282 2 "RM" }{TEXT -1 3 "] [" } {TEXT 283 2 "CM" }{TEXT -1 21 "] is the following " }{TEXT 432 1 "( " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 433 3 " \327 " }{XPPEDIT 18 0 " 1" "6#\"\"\"" }{TEXT 434 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "`RM CM` := multiply(RM, CM) : RM* CM = matrix(`RM CM`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#RMG\"\" \"%#CMGF&-%'matrixG6#7#7#,(*&&%#rmG6#\"#6F&&%#cmGF1F&F&*&&F06#\"#7F&&F 46#\"#@F&F&*&&F06#\"#8F&&F46#\"#JF&F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "To convert the re sultant matrix to the corresponding " }{TEXT 435 6 "scalar" }{TEXT -1 68 " (number), extract the matrix element using the subscript nota tion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "`RM CM` := `RM CM` [1,1] : RM*CM = `RM CM` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#RM G\"\"\"%#CMGF&,(*&&%#rmG6#\"#6F&&%#cmGF,F&F&*&&F+6#\"#7F&&F/6#\"#@F&F& *&&F+6#\"#8F&&F/6#\"#JF&F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "This matrix multiplication may be displayed in \"like-in-a-book\" form, namely" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "matrix(RM) * matrix(CM) = `RM CM` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%'matrixG6#7#7%&%#rmG6#\"#6&F+6#\"#7&F+6#\" #8\"\"\"-F&6#7%7#&%#cmGF,7#&F:6#\"#@7#&F:6#\"#JF4,(*&F*F4F9F4F4*&F.F4F " 0 "" {MPLTEXT 1 0 103 "RM := matrix(1, 3, [1, -1, 2]) : \+ CM := matrix(3, 1, [2, 1, -3]) : RM=matrix(RM) ; CM=matrix(CM) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#RMG-%'matrixG6#7#7%\"\"\"!\"\" \"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#CMG-%'matrixG6#7%7#\"\"#7# \"\"\"7#!\"$" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The product [" }{TEXT 274 2 "RM" }{TEXT -1 3 " ] [" }{TEXT 275 2 "CM" }{TEXT -1 21 "] is the following " }{TEXT 438 1 "(" }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 439 3 " \327 " } {XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 440 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "`RM CM` := multiply(RM, C M) : RM*CM = matrix(`RM CM`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/* &%#RMG\"\"\"%#CMGF&-%'matrixG6#7#7#!\"&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Conversion to sca lar yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "`RM CM` := `R M CM`[1,1] : RM*CM = `RM CM` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/* &%#RMG\"\"\"%#CMGF&!\"&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "or, in \"like-in-a-book\" form," } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "matrix(RM) * matrix(CM) = \+ `RM CM` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%'matrixG6#7#7%\"\"\" !\"\"\"\"#F*-F&6#7%7#F,7#F*7#!\"$F*!\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 294 1 "B" }{TEXT -1 2 ". " }{TEXT 270 50 "Multiplication of a column matrix and a row matrix " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "According to the multiplication conformability rule, the \+ product of an " }{TEXT 388 1 "(" }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 389 3 " \327 " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 390 1 ")" }{TEXT -1 2 " " }{TEXT 260 13 "column matrix" }{TEXT -1 3 " [" }{TEXT 265 2 "CM" }{TEXT -1 9 "] and a " }{TEXT 391 1 "(" }{XPPEDIT 18 0 "1" "6# \"\"\"" }{TEXT 392 3 " \327 " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 393 1 ")" }{TEXT -1 2 " " }{TEXT 262 10 "row matrix" }{TEXT -1 3 " [" } {TEXT 266 2 "RM" }{TEXT -1 122 "] is possible only, if the number of r ows in the column matrix is equal to the number of columns in the row \+ matrix, i.e. " }{XPPEDIT 18 0 "m=n" "6#/%\"mG%\"nG" }{TEXT -1 28 ". \+ The product matrix is a " }{TEXT 261 13 "square matrix" }{TEXT -1 12 " of order " }{TEXT 394 1 "(" }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 395 3 " \327 " }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 396 1 ")" }{TEXT -1 6 " or " }{TEXT 397 1 "(" }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 398 3 " \327 " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 399 1 ")" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "For example, consider the same " }{TEXT 400 1 "(" } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 401 3 " \327 " }{XPPEDIT 18 0 "1" " 6#\"\"\"" }{TEXT 402 1 ")" }{TEXT -1 17 " column matrix [" }{TEXT 284 2 "CM" }{TEXT -1 11 "] and the " }{TEXT 403 1 "(" }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 404 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 405 1 ")" }{TEXT -1 14 " row matrix [" }{TEXT 285 2 "RM" } {TEXT -1 48 "] with symbolic elements, as defined in Section " }{TEXT 341 1 "A" }{TEXT -1 14 " of this Unit." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "CM := matrix(3, 1, [cm[11], cm[21], cm[31]]) : RM : = matrix(1, 3, [rm[11], rm[12], rm[13]]) :" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The product [" } {TEXT 276 2 "CM" }{TEXT -1 3 "] [" }{TEXT 277 2 "RM" }{TEXT -1 21 "] \+ is the following " }{TEXT 406 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 407 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 408 1 ")" } {TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "`C M RM` := multiply(CM, RM) : `CM RM` = matrix(`CM RM`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&CM~RMG-%'matrixG6#7%7%*&&%#rmG6#\"#6\"\"\"&% #cmGF-F/*&F0F/&F,6#\"#7F/*&F0F/&F,6#\"#8F/7%*&&F16#\"#@F/F+F/*&F3F/F " 0 "" {MPLTEXT 1 0 43 "matrix(CM) * matrix(RM) = matrix(`CM RM`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%'matrixG6 #7#7%&%#rmG6#\"#6&F+6#\"#7&F+6#\"#8\"\"\"-F&6#7%7#&%#cmGF,7#&F:6#\"#@7 #&F:6#\"#JF4-F&6#7%7%*&F*F4F9F4*&F9F4F.F4*&F9F4F1F47%*&F " 0 "" {MPLTEXT 1 0 103 "CM := matrix(3, 1, [ 2, 1, -3]) : RM := matrix(1, 3, [1, -1, 2]) : CM=matrix(CM) ; RM =matrix(RM) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#CMG-%'matrixG6#7%7 #\"\"#7#\"\"\"7#!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#RMG-%'matri xG6#7#7%\"\"\"!\"\"\"\"#" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The product [" }{TEXT 289 2 "CM" }{TEXT -1 3 "] [" }{TEXT 290 2 "RM" }{TEXT -1 21 "] is the following \+ " }{TEXT 415 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 416 3 " \327 \+ " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 417 1 ")" }{TEXT -1 9 " matrix: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "`CM RM` := multiply(CM, RM) : `CM RM` = matrix(`CM RM`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%&CM~RMG-%'matrixG6#7%7%\"\"#!\"#\"\"%7%\"\"\"!\"\"F*7%!\"$\"\"$!\" '" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "or, in \"like-in-a-book\" form," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "matrix(CM) * matrix(RM) = matrix(`CM RM`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%'matrixG6#7#7%\"\"\"!\"\"\"\"#F*- F&6#7%7#F,7#F*7#!\"$F*-F&6#7%7%F,!\"#\"\"%F)7%F3\"\"$!\"'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 425 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Proceed to Unit (6) for \"" }{TEXT 423 69 "Multiplic ation of a multi-row multi-column matrix and a column matrix" }{TEXT -1 2 "\"." }}}{EXCHG {PARA 258 "" 0 "" {TEXT 424 67 "----------------- --------------------------------------------------" }}}}{MARK "146 0 0 " 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }