sol42 - Partial Solution Set, Leon Section 4.2 4.2.2a...

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Unformatted text preview: Partial Solution Set, Leon Section 4.2 4.2.2a Define L : R 3 R 2 by L ( ( x 1 ,x 2 ,x 3 ) T ) = ( x 1 + x 2 , 0) T . Find a matrix A such that L ( x ) = A x for every x R 3 . Solution : We take as the columns of A the images under L of the standard basis vectors from R 2 , obtaining A = 1 1 0 0 0 0 . 4.2.4 L : R 3 R 3 is given by L( x ) = (2 x 1- x 2- x 3 , 2 x 2- x 1- x 3 , 2 x 3- x 1- x 2 ) T . The standard matrix representation of L is (again) the matrix A whose i th column is L ( e i ), i.e., A = 2- 1- 1- 1 2- 1- 1- 1 2 . 4.2.5b L is the operator on R 2 whose effect is to first reflect a vector x about the x 1 axis and then rotate it counterclockwise through an angle of / 2. We are to find the standard matrix representation for L . This can be found by considering the effect of the transformation on the standard basis vectors: e 1 is unaffected by the reflection, and the subsequent rotation produces a copy of e 2 , while the reflection sends e 2 to- e 2 , and the subsequent rotation sends- e 2 to e 1 . The overall effect of L is reflection about the identity line. The matrix is A = 0 1 1 0 . Note that we can do this in a different (and simpler, if messier) way. It is not hard to show that the matrix representation of the composition of transformations is the product of the individual matrix representations. So let L = L 2 L 1 , where L 1 is the reflection and L 2 is the rotation. The standard matrix representations for L 1 and L 2 are A 1 = 1- 1 and A 2 =- 1 1 . Thus A =- 1 1 1- 1 = 0 1 1 0 . 4.2.6 We are given three vectors b 1 , b 2 , and b 3 . The linear transformation L , mapping R 2 to R 3 , is given by L ( x ) = x 1 b 1 + x 2 b 2 + ( x 1 + x 2 ) b 3 . The problem is to find the matrix A representing L with respect to the bases [ e 1 , e 2 ] and B = [ b 1 , b 2 , b 3 ]. The matrix is A= a 1 a 2 , where a i = [ L ( e i )] B . Thus to find a 1 we first compute L ( e 1 ) = b 1 + b 3 ; the coordinate vector of b 1 + b 3 with respect to the basis B is (1 , , 1) T . Similarly we find....
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sol42 - Partial Solution Set, Leon Section 4.2 4.2.2a...

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