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}{PSTYLE "_pstyle9" -1 209 1 {CSTYLE "" -1 -1 "T imes" 1 10 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {CSTYLE "_cstyle14" -1 217 "Times" 1 10 0 0 0 1 1 1 2 2 2 2 0 0 0 1 } {PSTYLE "_pstyle10" -1 210 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 201 "" 0 "" {TEXT 204 32 "A Quick Introduction to Linear A" }{TEXT 204 7 "lgebra " }}{PARA 201 "" 0 "" {TEXT 204 20 "fo r Math 374 at WMU." }}{PARA 202 "" 0 "" {TEXT 205 7 "Part II" }}{PARA 202 "" 0 "" {TEXT 205 11 "Jay Treiman" }}{PARA 203 "" 0 "" }{PARA 202 "" 0 "" {TEXT 205 8 "May 2004" }}{PARA 202 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 53 "The two worksheets in this set only cover the linear " } }{PARA 203 "" 0 "" {TEXT 206 56 "algebra in the syllibus for Math 272 \+ at WMU. It is not " }}{PARA 203 "" 0 "" {TEXT 206 58 "intended to giv e any more material or to give the student " }}{PARA 203 "" 0 "" {TEXT 206 65 "more than a basic introduction to the linear algebra in \+ Math 272." }}}{SECT 1 {PARA 204 "" 0 "" {HYPERLNK 207 "Restart" 2 "res tart" "" }{TEXT 208 14 " and load the " }{HYPERLNK 207 "LinearAlgebra" 2 "LinearAlgebra" "" }{TEXT 208 9 " package." }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 209 8 "restart;" }{MPLTEXT 1 209 21 "\nwith(LinearAlg ebra):" }}}}{SECT 1 {PARA 204 "" 0 "" {TEXT 208 12 "Determinants" }} {PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 56 "The determinant of \+ a matrix is a number that is formally" }}{PARA 203 "" 0 "" {TEXT 206 61 "defined thorugh a somewhat confusing formula. It is easy to " }} {PARA 203 "" 0 "" {TEXT 206 45 "calculate for 2 by 2 and 3 by three ma trices." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 38 "For a 2 \+ by 2 matrix the determinant is" }}{PARA 205 "" 0 "" {TEXT 210 3 " " }{XPPEDIT 18 0 "det(matrix([[a, b], [c, d]])) = a*d-b*c;" "6#/-%$detG6 #-%'matrixG6#7$7$%\"aG%\"bG7$%\"cG%\"dG,&*&F,\"\"\"F0F3F3*&F-F3F/F3!\" \"" }{TEXT 211 1 " " }{TEXT 210 1 "." }}{PARA 203 "" 0 "" }{PARA 203 " " 0 "" {TEXT 206 69 "The determinant for a 3 by 3 matrix can be writte n through a cofactor" }}{PARA 203 "" 0 "" {TEXT 206 16 "expansion. He re" }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 2 " " }{XPPEDIT 18 0 "det(matrix([[a[1,1], a[1,2], a[1,3]], [a[2,1], a[2,2], a[2,3]], \+ [a[3,1], a[3,2], a[3,3]]])) = a[1,1]*det(matrix([[a[2,2], a[2,3]], [a[ 3,2], a[3,3]]]))-a[1,2]*det(matrix([[a[2,1], a[2,3]], [a[3,1], a[3,3]] ]))+a[1,3]*det(matrix([[a[2,1], a[2,2]], [a[3,1], a[3,2]]]));" "6#/-%$ detG6#-%'matrixG6#7%7%&%\"aG6$\"\"\"F/&F-6$F/\"\"#&F-6$F/\"\"$7%&F-6$F 2F/&F-6$F2F2&F-6$F2F57%&F-6$F5F/&F-6$F5F2&F-6$F5F5,(*&F,F/-F%6#-F(6#7$ 7$F9F;7$F@FBF/F/*&F0F/-F%6#-F(6#7$7$F7F;7$F>FBF/!\"\"*&F3F/-F%6#-F(6#7 $7$F7F97$F>F@F/F/" }{TEXT 212 1 " " }{TEXT 206 1 "." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 60 "This can be written out as a formu la without determinants of" }}{PARA 203 "" 0 "" {TEXT 206 17 "smaller \+ matrices." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 2 " " } {XPPEDIT 18 0 "det(matrix([[a[1,1], a[1,2], a[1,3]], [a[2,1], a[2,2], \+ a[2,3]], [a[3,1], a[3,2], a[3,3]]])) = a[1,1]*a[2,2]*a[3,3]-a[1,1]*a[2 ,3]*a[3,2]-a[2,1]*a[1,2]*a[3,3]+a[2,1]*a[1,3]*a[3,2]+a[3,1]*a[1,2]*a[2 ,3]-a[3,1]*a[1,3]*a[2,2];" "6#/-%$detG6#-%'matrixG6#7%7%&%\"aG6$\"\"\" F/&F-6$F/\"\"#&F-6$F/\"\"$7%&F-6$F2F/&F-6$F2F2&F-6$F2F57%&F-6$F5F/&F-6 $F5F2&F-6$F5F5,.*(F,F/F9F/FBF/F/*(F,F/F;F/F@F/!\"\"*(F7F/F0F/FBF/FG*(F 7F/F3F/F@F/F/*(F>F/F0F/F;F/F/*(F>F/F3F/F9F/FG" }{TEXT 212 1 " " } {TEXT 206 1 "." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 52 "The determinants for 2 by 2 and 3 by 3 matrices are \+ " }}{PARA 203 "" 0 "" {TEXT 206 32 "easily computed using the Maple " }{HYPERLNK 213 "Determinant" 2 "LinearAlgebra,Determinant" "" }{TEXT 206 9 " command." }}{PARA 203 "" 0 "" }{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 209 27 "A := Matrix([[a,b],[c,d]]);" }{MPLTEXT 1 209 16 "\n Determinant(A);" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 209 31 "B := Matrix(3,3,(i,j)->b[i,j]);" }{MPLTEXT 1 209 24 "\nDet_B :=Determinant (B);" }}}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 55 "It is easi er to remember the cofactor expansion formula" }}{PARA 203 "" 0 "" {TEXT 206 32 "for a 3 by 3 determinant. Here " }{XPPEDIT 18 0 "B[i,j] ;" "6#&%\"BG6$%\"iG%\"jG" }{TEXT 212 1 " " }{TEXT 206 19 " dnotes the \+ matrix " }}{PARA 203 "" 0 "" {TEXT 206 26 "obtained by deleting the " }{XPPEDIT 18 0 "i;" "6#%\"iG" }{TEXT 212 1 " " }{TEXT 206 11 "th row \+ and " }{XPPEDIT 18 0 "j;" "6#%\"jG" }{TEXT 212 1 " " }{TEXT 206 16 "th column. Then" }}{PARA 203 "" 0 "" {TEXT 206 2 " " }{XPPEDIT 18 0 "d et(B) = sum((-1)^(i+j)*b[1,j]*det(B[i, j]), j = (1 .. 3))" "6#/-I$detG 6\"6#I\"BGF&-I$sumGF&6$*(),$\"\"\"!\"\",&I\"iGF&F/I\"jGF&F/F/&I\"bGF&6 $F/F3F/-F%6#&F(6$F2F3F//F3;F/\"\"$" }{TEXT 212 1 " " }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 6 "where " }{XPPEDIT 18 0 "i;" "6#%\"i G" }{TEXT 212 1 " " }{TEXT 206 21 " represents a row of " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT 212 1 " " }{TEXT 206 26 ". The same foruml a holds " }}{PARA 203 "" 0 "" {TEXT 206 27 "for a column expansion for " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT 212 1 " " }{TEXT 206 25 ". Th is formula holds for" }}{PARA 203 "" 0 "" {TEXT 206 10 "arbitrary " } {XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT 212 1 " " }{TEXT 206 4 " by " } {XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT 212 1 " " }{TEXT 206 10 " matrices ." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 57 "There is also \+ a way of calculating determinants of 2 by 2" }}{PARA 203 "" 0 "" {TEXT 206 53 " and 3 by 3 matrices using the sum of products along " } }{PARA 203 "" 0 "" {TEXT 206 56 "diagonals going up and down. See a l inear algebra text " }}{PARA 203 "" 0 "" {TEXT 206 19 "for these formu las." }}{PARA 203 "" 0 "" }{SECT 1 {PARA 206 "" 0 "" {TEXT 214 41 "The inverse of a 2 by 2 and determinants." }}{PARA 203 "" 0 "" {TEXT 206 41 "There is a formula for the inverse of an " }{XPPEDIT 18 0 "n;" "6# %\"nG" }{TEXT 212 1 " " }{TEXT 206 4 " by " }{XPPEDIT 18 0 "n;" "6#%\" nG" }{TEXT 212 1 " " }{TEXT 206 7 " matrix" }}{PARA 203 "" 0 "" {TEXT 206 52 "that uses determinants. It is relatively simple for" }}{PARA 203 "" 0 "" {TEXT 206 38 "a 2 by 2 matrix. In this case one has" }} {PARA 203 "" 0 "" {TEXT 206 2 " " }{XPPEDIT 18 0 "B^(-1) = 1/det(B);" "6#/)%\"BG,$\"\"\"!\"\"*&F'F'-%$detG6#F%F(" }{TEXT 212 1 " " }{TEXT 206 2 " " }{XPPEDIT 18 0 "MATRIX([[b[2,2], -b[1,2]], [-b[2,1], b[1,1] ]]);" "6#-%'MATRIXG6#7$7$&%\"bG6$\"\"#F+,$&F)6$\"\"\"F+!\"\"7$,$&F)6$F +F/F0&F)6$F/F/" }{TEXT 212 1 " " }}}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 75 "There are several special cases when it is easy to ca lculate a determinant." }}{PARA 203 "" 0 "" {TEXT 206 78 "If a matrix \+ is diagonal, i.e. all entries off the main diagonal are zero, then" }} {PARA 203 "" 0 "" {TEXT 206 71 "the determinant is the product of the \+ diagonal elements. Similarly, if" }}{PARA 203 "" 0 "" {TEXT 206 73 "a matrix is upper (lower) triangular, i.e. all elements below (above) t he" }}{PARA 203 "" 0 "" {TEXT 206 65 "main diagonal are zero, the the \+ determinant is the product of the" }}{PARA 203 "" 0 "" {TEXT 206 32 "d iagonal elements of the matrix." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 " " {TEXT 206 23 "Here are some examples." }}{PARA 203 "" 0 "" }{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 209 31 "C := DiagonalMatrix([a,b,c,d]); " }{MPLTEXT 1 209 25 "\ndet_C := Determinant(C);" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 209 41 "E := Matrix([[-2,3,4],[0,5,1],[0,0,-3]]); " }{MPLTEXT 1 209 25 "\ndet_E := Determinant(E);" }}}{PARA 203 "" 0 "" }{SECT 1 {PARA 206 "" 0 "" {TEXT 214 9 "Exercises" }}{PARA 203 "" 0 " " }{SECT 1 {PARA 207 "" 0 "" {TEXT 215 1 "1" }}{PARA 203 "" 0 "" {TEXT 206 24 "Find the determinant of " }{XPPMATH 200 "6#-%'MATRIXG6#7 $7$\"\"\"\"\"#7$F)F(" }{TEXT 206 9 " by hand." }}{EXCHG {PARA 203 "> " 0 "" }}}{SECT 1 {PARA 207 "" 0 "" {TEXT 215 1 "2" }}{PARA 203 "" 0 "" {TEXT 206 24 "Find the determinant of " }{XPPMATH 200 "6#-%'MATRIXG6# 7%7%\"\"\"\"\"#\"\"$7%F*F)F(7%F(F(F(" }{TEXT 206 25 " by hand and usin g Maple." }}{EXCHG {PARA 203 "> " 0 "" }}}{SECT 1 {PARA 207 "" 0 "" {TEXT 215 1 "3" }}{PARA 203 "" 0 "" {TEXT 206 24 "Find the determinant of " }{XPPMATH 200 "6#-%'MATRIXG6#7&7&\"\"\"\"\"!F)F)7&\"\"#F)F(F)7&F (F(\"\"$F)7&F)F)\"\"%F/" }{TEXT 206 25 " by hand and using Maple." }} {EXCHG {PARA 203 "> " 0 "" }}}{SECT 1 {PARA 207 "" 0 "" {TEXT 215 1 "4 " }}{PARA 203 "" 0 "" {TEXT 206 25 "Calculate the inverse of " } {XPPMATH 200 "6#-%'MATRIXG6#7$7$\"\"\"\"\"#7$F)F(" }{TEXT 206 25 " by \+ hand and using Maple." }}{EXCHG {PARA 203 "> " 0 "" }}}}}{SECT 1 {PARA 204 "" 0 "" {TEXT 208 22 "Linear Transformations" }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 73 "A linear transformation is a sp ecial type of function. It satisfies the " }}{PARA 203 "" 0 "" {TEXT 206 25 "following two properties." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 6 "(1) " }{XPPEDIT 18 0 "T(x+y) = T(x)+T(y);" "6#/-%\" TG6#,&%\"xG\"\"\"%\"yGF),&-F%6#F(F)-F%6#F*F)" }{TEXT 212 1 " " }{TEXT 206 1 "," }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 3 "and" }} {PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 6 "(2) " }{XPPEDIT 18 0 "T(alpha*x) = alpha*T(x);" "6#/-%\"TG6#*&%&alphaG\"\"\"%\"xGF)*&F (F)-F%6#F*F)" }{TEXT 212 1 " " }{TEXT 206 2 ". " }}{PARA 203 "" 0 "" } {PARA 203 "" 0 "" {TEXT 206 66 "There are quite a few functions that s atisfy these two properites." }}{PARA 203 "" 0 "" {TEXT 206 64 "The mo st important ones for multivariate calculus are those from" }}{PARA 203 "" 0 "" {TEXT 206 1 " " }{XPPEDIT 18 0 "R^n;" "6#)%\"RG%\"nG" } {TEXT 212 1 " " }{TEXT 206 4 " to " }{XPPEDIT 18 0 "R^m;" "6#)%\"RG%\" mG" }{TEXT 212 1 " " }{TEXT 206 60 ". Every one of these linear tran sformations can be written" }}{PARA 203 "" 0 "" {TEXT 206 11 "in the f orm" }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 1 " " }{XPPEDIT 18 0 "T(y)[i] = sum(a[i,j]*y[j],j = 1 .. n);" "6#/&-%\"TG6#%\"yG6#%\"i G-%$sumG6$*&&%\"aG6$F*%\"jG\"\"\"&F(6#F2F3/F2;F3%\"nG" }{TEXT 212 1 " \+ " }{TEXT 206 1 "." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 45 "The left hand side of this expression is the " }{XPPEDIT 18 0 "i;" "6#%\"iG" }{TEXT 212 1 " " }{TEXT 206 16 "th component of " } {XPPEDIT 18 0 "T(y);" "6#-%\"TG6#%\"yG" }{TEXT 212 1 " " }{TEXT 206 1 "." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 13 "If one takes \+ " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT 212 1 " " }{TEXT 206 11 " to be the " }{XPPEDIT 2 0 "m" "6#I\"mG6\"" }{TEXT 212 1 " " }{TEXT -1 4 " b y " }{XPPEDIT 2 0 "n" "6#I\"nG6\"" }{TEXT 212 1 " " }{TEXT -1 1 " " } {TEXT 206 20 "matrix with entries " }{XPPEDIT 18 0 "a[i,j];" "6#&%\"aG 6$%\"iG%\"jG" }{TEXT 212 1 " " }{TEXT 206 20 ", then one can write" }} {PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 1 " " }{XPPEDIT 18 0 "T (y) = A*y;" "6#/-%\"TG6#%\"yG*&%\"AG\"\"\"F'F*" }{TEXT 212 1 " " } {TEXT 206 1 "." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 40 "I n fact, any linear transformation from " }{XPPEDIT 18 0 "R^n;" "6#)%\" RG%\"nG" }{TEXT 212 1 " " }{TEXT 206 4 " to " }{XPPEDIT 18 0 "R^m;" "6 #)%\"RG%\"mG" }{TEXT 212 1 " " }{TEXT 206 25 " can be written this way ." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 31 "For example, o ne can check that" }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 1 " " }{XPPEDIT 18 0 "T([y[1], y[2]]) = [2*y[1]-y[2], -y[1]+6*y[2]];" "6 #/-%\"TG6#7$&%\"yG6#\"\"\"&F)6#\"\"#7$,&*&F.F+F(F+F+F,!\"\",&F(F2*&\" \"'F+F,F+F+" }{TEXT 212 1 " " }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 26 "can also be represented as" }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {XPPEDIT 18 0 "T(y) = matrix([[2, -1], [-1, 6]])*y;" "6#/- %\"TG6#%\"yG*&-%'matrixG6#7$7$\"\"#,$\"\"\"!\"\"7$F/\"\"'F0F'F0" } {TEXT 212 1 " " }{TEXT 206 1 "." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 " " {TEXT 206 24 "Here is a definition of " }{XPPEDIT 18 0 "T;" "6#%\"TG " }{TEXT 212 1 " " }{TEXT 206 35 " in Maple. Note how one can define" }}{PARA 208 "" 0 "" {XPPEDIT 2 0 "T" "6#I\"TG6\"" }{TEXT 2 1 " " } {TEXT 200 18 " as a function of " }{XPPEDIT 2 0 "y" "6#I\"yG6\"" } {TEXT 211 1 " " }{TEXT 200 1 "." }{TEXT 2 1 " " }}{PARA 203 "" 0 "" } {EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 209 21 "A := <<2,-1>|<-1,6>>;" } {MPLTEXT 1 209 25 "\ny1 := Vector(2,i->y[i]);" }{MPLTEXT 1 209 24 "\nT := y-> Multiply(A,y):" }{MPLTEXT 1 209 18 "\nT_of_y1 := T(y1);" }}} {PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 73 "This makes it easy \+ to see that the composition of linear transformations " }}{PARA 203 "" 0 "" {TEXT 206 36 "translates to matrix multiplication." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 39 "Here is a second linear trans formation," }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 1 " " } {XPPEDIT 18 0 "S(x) = matrix([[1, -1], [0, 3]]);" "6#/-%\"SG6#%\"xG-%' matrixG6#7$7$\"\"\",$F-!\"\"7$\"\"!\"\"$" }{TEXT 212 1 " " }{TEXT 206 1 "." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" }{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 209 20 "B := <<1,-1>|<0,3>>;" }{MPLTEXT 1 209 25 "\nx1 : = Vector(2,i->x[i]);" }{MPLTEXT 1 209 24 "\nS := x-> Multiply(B,x):" } {MPLTEXT 1 209 18 "\nS_of_x1 := S(x1);" }}}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 47 "Now we can compose the two transformations a nd " }}{PARA 203 "" 0 "" {TEXT 206 50 "compare that with the product o f the two matrices." }}{PARA 203 "" 0 "" }{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 209 25 "S_of_T_of_y1 := S(T(y1));" }{MPLTEXT 1 209 18 "\nB_ times_A := B.A;" }{MPLTEXT 1 209 18 "\nB_A_y1 := B.A.y1;" }}}{EXCHG {PARA 203 "> " 0 "" }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 72 "It is clear that these are equal. If you have trouble remembering that " }}{PARA 203 "" 0 "" {TEXT 206 59 "matrix multiplication does n ot commute, remember that it is" }}{PARA 203 "" 0 "" {TEXT 206 57 "equ ivalent to the composition of linear transformations.." }}{PARA 203 "" 0 "" }{SECT 1 {PARA 206 "" 0 "" {TEXT 214 9 "Exercises" }}{SECT 1 {PARA 207 "" 0 "" {TEXT 215 1 "5" }}{PARA 203 "" 0 "" {TEXT 216 41 "Ar e the following linear transformations?" }}{PARA 203 "" 0 "" }{SECT 1 {PARA 209 "" 0 "" {TEXT 217 1 "a" }}{PARA 203 "" 0 "" {TEXT 216 35 "T( x, y, z) = [2*x-3*y+z, 4*x-y-6*z]" }}{EXCHG {PARA 203 "> " 0 "" }}} {SECT 1 {PARA 209 "" 0 "" {TEXT 217 1 "b" }}{PARA 203 "" 0 "" {TEXT 216 26 "S(z, w) = [3*w-z, w+3*z-2]" }}{EXCHG {PARA 203 "> " 0 "" }}} {SECT 1 {PARA 209 "" 0 "" {TEXT 217 1 "c" }}{PARA 203 "" 0 "" {TEXT 216 32 "H(x, y) = [x^2+y, sin(x)+cos(y)]" }}{EXCHG {PARA 203 "> " 0 "" }}}}{SECT 1 {PARA 207 "" 0 "" {TEXT 215 1 "6" }}{PARA 203 "" 0 "" {TEXT 206 54 "Find the matrix representing the linear transformation" }}{PARA 203 "" 0 "" {TEXT 206 3 " " }{XPPEDIT 18 0 "T(x,y,z) = [x-y+ 3*z, 2*x-2*y+z];" "6#/-%\"TG6%%\"xG%\"yG%\"zG7$,(F'\"\"\"F(!\"\"*&\"\" $F,F)F,F,,(*&\"\"#F,F'F,F,*&F2F,F(F,F-F)F," }{TEXT 212 1 " " }{TEXT 212 1 "." }}{EXCHG {PARA 203 "> " 0 "" }}}{SECT 1 {PARA 207 "" 0 "" {TEXT 215 1 "7" }}{PARA 203 "" 0 "" {TEXT 206 54 "Find the matrix repr esenting the linear transformation" }}{PARA 203 "" 0 "" {TEXT 206 3 " \+ " }{XPPEDIT 18 0 "T(x,y,z) = [x, y, z];" "6#/-%\"TG6%%\"xG%\"yG%\"zG 7%F'F(F)" }{TEXT 212 1 " " }{TEXT 212 1 "." }}{EXCHG {PARA 203 "> " 0 "" }}}}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" }{SECT 1 {PARA 206 "" 0 "" {TEXT 214 13 "Invertability" }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 24 "A linear transformation " }{XPPEDIT 18 0 "T;" "6#%\"TG" }{TEXT 212 1 " " }{TEXT 206 23 " is invertable if there" }}{PARA 203 " " 0 "" {TEXT 206 27 "is a linear transformation " }{XPPEDIT 18 0 "S;" "6#%\"SG" }{TEXT 212 1 " " }{TEXT 206 10 " such that" }}{PARA 203 "" 0 "" {TEXT 206 3 " " }{XPPEDIT 18 0 "S(T(x)) = x;" "6#/-%\"SG6#-%\"TG 6#%\"xGF*" }{TEXT 212 1 " " }{TEXT 206 7 " and " }{XPPEDIT 18 0 "T(S (x)) = x;" "6#/-%\"TG6#-%\"SG6#%\"xGF*" }{TEXT 212 1 " " }{TEXT 206 1 "." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 33 "If the matric es corresponding to " }{XPPEDIT 18 0 "T;" "6#%\"TG" }{TEXT 212 1 " " } {TEXT 206 5 " and " }{XPPEDIT 18 0 "S;" "6#%\"SG" }{TEXT 212 1 " " } {TEXT 206 4 " are" }}{PARA 203 "" 0 "" {TEXT 206 1 " " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT 212 1 " " }{TEXT 206 5 " and " }{XPPEDIT 18 0 "B ;" "6#%\"BG" }{TEXT 212 1 " " }{TEXT 206 17 ", this means that" }} {PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 206 2 " " }{XPPEDIT 18 0 " A*B*x = x;" "6#/*(%\"AG\"\"\"%\"BGF&%\"xGF&F(" }{TEXT 212 1 " " } {TEXT 206 7 " and " }{XPPEDIT 18 0 "B*A*x = x;" "6#/*(%\"BG\"\"\"%\" AGF&%\"xGF&F(" }{TEXT 212 1 " " }{TEXT 206 1 "." }}{PARA 203 "" 0 "" } {PARA 203 "" 0 "" {TEXT 206 18 "This implies that " }{XPPEDIT 18 0 "A; " "6#%\"AG" }{TEXT 212 1 " " }{TEXT 206 5 " and " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT 212 1 " " }{TEXT 206 20 " are inverses. Thus" }} {PARA 203 "" 0 "" {TEXT 206 25 "a linear transoformation " }{XPPEDIT 18 0 "T;" "6#%\"TG" }{TEXT 212 1 " " }{TEXT 206 22 " is invertable if \+ and " }}{PARA 203 "" 0 "" {TEXT 206 46 "only if the corrsponding matri x is invertable." }}{PARA 203 "" 0 "" }{SECT 1 {PARA 207 "" 0 "" {TEXT 215 9 "Exercises" }}{PARA 203 "" 0 "" }{SECT 1 {PARA 209 "" 0 "" {TEXT 217 1 "8" }}{PARA 203 "" 0 "" {TEXT 206 34 "Check if the linear transformation" }}{PARA 203 "" 0 "" {TEXT 206 2 " " }{XPPEDIT 18 0 " T(x,y,z) = [x-y+z, 2*x-3*y+2*z];" "6#/-%\"TG6%%\"xG%\"yG%\"zG7$,(F'\" \"\"F(!\"\"F)F,,(*&\"\"#F,F'F,F,*&\"\"$F,F(F,F-*&F0F,F)F,F," }{TEXT 212 1 " " }}{PARA 203 "" 0 "" {TEXT 206 41 "is invertable. If it is, f ind the inverse" }}{PARA 203 "" 0 "" {TEXT 206 37 "transformation by f inding the inverse" }}{PARA 203 "" 0 "" {TEXT 206 27 "of the matrix re presenting " }{XPPEDIT 18 0 "T;" "6#%\"TG" }{TEXT 212 1 " " }{TEXT 212 1 "." }}{EXCHG {PARA 203 "> " 0 "" }}}{SECT 1 {PARA 209 "" 0 "" {TEXT 217 1 "9" }}{PARA 203 "" 0 "" {TEXT 206 34 "Check if the linear \+ transformation" }}{PARA 203 "" 0 "" {TEXT 206 2 " " }{XPPEDIT 18 0 "T (x,y,z) = [x-y+z, 2*x-3*y+2*z, 6*x-3*y+2*z];" "6#/-%\"TG6%%\"xG%\"yG% \"zG7%,(F'\"\"\"F(!\"\"F)F,,(*&\"\"#F,F'F,F,*&\"\"$F,F(F,F-*&F0F,F)F,F ,,(*&\"\"'F,F'F,F,F1F-F3F," }{TEXT 212 1 " " }}{PARA 203 "" 0 "" {TEXT 206 41 "is invertable. If it is, find the inverse" }}{PARA 203 " " 0 "" {TEXT 206 37 "transformation by finding the inverse" }}{PARA 203 "" 0 "" {TEXT 206 27 "of the matrix representing " }{XPPEDIT 18 0 "T;" "6#%\"TG" }{TEXT 212 1 " " }{TEXT 212 1 "." }}{EXCHG {PARA 203 "> " 0 "" }}}}}}{PARA 210 "" 0 "" }}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }