Linear algebra problem

The following linear transformations problem was confusing me, so any help would be really appreciated!

Let L: R^3 -> R^3 be given by L([x, y, z]) = [-4y - 13z, -6x + 5y + 6z, 2x - 2y - 3z].

What is the matrix for L with respect to the basis
B = ([-1, 6, 2], [3, 4, -1], [-1, -3, 1])?

buckaroobill said:

The following linear transformations problem was confusing me, so any help would be really appreciated!

Let L: R^3 -> R^3 be given by L([x, y, z]) = [-4y - 13z, -6x + 5y + 6z, 2x - 2y - 3z].

What is the matrix for L with respect to the basis
B = ([-1, 6, 2], [3, 4, -1], [-1, -3, 1])?

Click to expand...

To find \(\displaystyle \[L\]_B^B\) (or \(\displaystyle _B \[L\] _B\) depending on notation) set up the matrix below as column vectors. (Even if you are given row vectors you do this.)

\(\displaystyle \L \[ B_1 \,\, B_2 \,\, B_3 \,\, \| \,\, L\(B_1\) \,\, L\(B_2\) \,\, L\(B_3\) \]\)

Then row-reduce the LHS matrix to the idenity matrix, while making sure to perform the same operations to the augmented matrix. The matrix on the RHS will be your trasition matrix.