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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.

NotAStudentForReal
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  • I can't tell what you were doing after you found $A$, but here's what you should have been doing. Either $(1)$ find the representation of the $L(u_1)$ and $L(u_2)$ wrt the basis $U={u_1, u_2}$ and then find the matrix which takes $[u_1]_U = \pmatrix{1 \ 0}$ to $[L(u_1)]_U$ and the same for $u_2$ OR $(2)$ find the change of basis matrix $P$ from $U$ to the standard basis and then your matrix $B$ will be $B=P^{-1}AP$. Use either method to find $B$. –  Oct 15 '15 at 00:40
  • Note however that there's a shortcut with that second method (because this transformation happens to be a particularly easy one). Just look at the equation $B=P^{-1}AP$. If $A=-I$ (and it does, doesn't it?), do we really need to find $P$ to figure out $B$? ;) Let me know if you need any more help. –  Oct 15 '15 at 00:44
  • Isn't finding the representation of the linear operator with respect to U just plugging u1 and u2 into the linear operator...? That was what I was told to do, though apparently that's not the case. How would I find the change of basis matrix P from U? – NotAStudentForReal Oct 15 '15 at 00:51
  • So... you don't want to use the shortcut? –  Oct 15 '15 at 00:57
  • Is B just -I? I'm more interested in learning the steps so I can apply this to other problems. – NotAStudentForReal Oct 15 '15 at 00:59
  • The change of basis matrix $P$ will just be $\pmatrix{[u_1]{\mathcal E} & [u_2]{\mathcal E}}$ where $\mathcal E$ is the standard basis. –  Oct 15 '15 at 00:59
  • See if you can understand this answer that I gave the other day. It's the same type of problem (change of basis). If there's a particular part you don't understand, let me know and I'll write up a solution. But try to go through my answer to that question first. –  Oct 15 '15 at 01:02
  • Yes, I was able to follow it! That's actually a really thorough walkthrough, thank you very much! I was also able to correctly complete the rest of the questions in my problem set too. – NotAStudentForReal Oct 15 '15 at 01:32

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