Question

Please answer the following. Thank you.

(1 point) Let A--5-5-5 5 |. Find basis for the kernal and image of the linear transformation T defined by T(刃 L-5-1 5, Kernel basis: Image basis: To enter a basis into WeBWorK, place the entries of each vector inside of brackets, and enter a list of these vectors, separated by commas. For instance, if your basis is2 1 I&, then you would enter [1,2,3],[1,1,1] into the answer blank. 3] L1

Answer 1

A = [-5 -5 -5 -7 -5 -1 -7 -5 -1 5 5 5] R_2 rightarrow R_2 - R_1 R_3 rightarrow R_3 - R_1 = > [-5 0 0 -7 2 6 -7 2 6 5 0 0] R_1 rightarrow R_1 + 7/2 R_2 R_3 rightarrow R-3 - 3 R_2 = > [-5 0 0 0 2 0 0 2 0 5 0 0] For Null space: -5a + 5d = 0 = > a = d 2b + 2c = 0 = > b = -c Therefore (a, b, c, d) = (d, -c, c, d) = (d, 0, 0, d) + (0, -c, c, 0) = d(1, 0, 0, 1) + c(0, -1, 1, 0) Kernel basis = {(1, 0, 0, 1), (0, -1, 1, 0)} Image basis = {(-5, 0, 0, 5), (0, 2, 2, 0)}