So I've been going through these problems and I got the first one in the set correct, but the second two wrong. I don't know why-- my arithmetic looks fine and the steps are pretty similar.
The question is:
For each of the following linear operations L on $\Bbb R^2$, determine the matrix A representing L with respect to {e1, e2} and the matrix B representing L with respect to {u1 = (1, 1)^T, u2 = (-1, 1)^T).
b) L(x) = -x
This is just applying a -1 to x, right?
$$A = \pmatrix{-1 & 0 \\ 0 & -1}$$
$L(u_1) = (-1, -1)^T, \quad L(u_2) = (1, -1)^T$
Inverse of u1, u2:
$$B = \pmatrix{-1/2 & -1/2 \\ 1/2 & -1/2}$$
Then I do U^-1 * L(u1, u2) and I obtain B:
$$B = \pmatrix{1 & 0 \\ 0 & 1}$$
However, B is wrong. The answer is:
$$B = \pmatrix{-1 & 0 \\ 0 & -1}$$
Can someone explain why this is? I'm sure I'm following the book's steps correctly.