7331513 - Now, =-=- - --=- -S1 AB 1e1...

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Solution: The standard matrix of a linear operator L is the matrix S representing L with respect to standard base = , , , = , , , = , , e1 1 0 0 e2 0 1 0 e3 0 0 1 and the base B given by: ( ) L ei B a) Standard matrix, = - S 13 87 and = B 1124 Let the matrix of linear operator L be: = A a11a12a21a22 We have, ( ) = - L ei B AB 1ei - = - - B 1 2 12 112 Now, = - = - - = - S1 AB 1e1 a11a12a21a222 12 11210 1 8 - = 2a11 a12 1 ........... [1] - =- 2a21 a12 8 ......... [2] Similarly, = - = - - = S2 AB 1e2 a11a12a21a222 12 11201 37 - + = a112 a122 3 .......... [3] - + = a212 a122 7 ....... [4] = , = , = , = a11 7 a12 13 a21 6 a22 20 Hence, required A matrix is = A 713620 b) Standard matrix, =- - - - - - S 6 29 5 17 7 210 and = B 111103112
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Let the matrix of linear operator L be: = A a11a12a13a21a22a23a31a32a33 We have, ( ) = - L ei B AB 1ei - = - - - - B 1 31 3 1 12 101
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Unformatted text preview: Now, =-=- - --=- -S1 AB 1e1 a11a12a13a21a22a23a31a32a3331 3 1 12 101100 6-5 7--=-3a11 a12 a13 6 ........... [1]--=- 3a21 a22 a23 5 .......... [2]--=-3a31 a32 a33 7 .......... [3] Similarly, =-=- - --S2 AB 1e2 a11a12a13a21a22a23a31a32a3331 3 1 12 =- - -101010 2 1 2-=-a11 a12 2 ......... [4]-=-a21 a22 1 ........... [5]-=-a31 a32 2 ........... [6] Similarly, =-=- - --S3 AB 1e3 a11a12a13a21a22a23a31a32a3331 3 1 12 = 101001 9710-+ + = 3a11 2a12 a13 9 ........... [7]-+ + = 3a21 2a22 a23 7 .......... [8]-+ + = 3a31 2a32 a33 10 ......... [9] Solving above 9 equations, we get: = , = , = , = , = , = , = , = , = a11 1 a21 1 a31 1 a12 3 a22 2 a32 3 a13 6 a23 6 a33 7 Hence, required A matrix is = A 136126137...
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7331513 - Now, =-=- - --=- -S1 AB 1e1...

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