Math 2660 Test 2, 9AM 1. 3 9 1 0 17 A := 1 3 2 1 8 . 0 0 1 2 0 Find bases B and B for the row-space U of A and for U , its orthogonal complement, also called the null space of A. Also find a basis C for the column-space of A. 2. Suppose v 1 := (1, 1, 2), v 2 := (0, 2, 4), v 3 := (0, 1, 3) is a second basis V for R3 , in addition to the standard basis E. If xE := (2, 0, 8), find xV . 3. Suppose L(x)E = xE AEE with AEE 5 1 2 0 1. := 0 1 1 1 For V the second basis in problem 2 above, if xV := (6, 0, 2), find xE , L(x)E , and L(x)V . 4. For each of the following L that is a linear transformation find a matrix A such that L(x) = xA: (a) L : R3 R2 , L((x1 , x2 , x3 )) := (2x1 + x2 4x3 , x2 6x3 ); (b) L : R3 R3 , L((x1 , x2 , x3 )) := (x1 + 2, x2 1, x3 + 3); (c) L : R2 R3 , L((x1 , x2 )) := (x1 x2 , 2x1 3x2 , 3x1 + 4x2 ). 5. Find the kernel and image of the linear transformation L : R4 R3 defined by L(x) := xA for 2 1 3 1 2 4 . A := 0 5 11 3 1 1 Find all the elements of the domain that have image (7, 1, 4). Find all the elements of the domain that have image (7, 1, 5). 1