Is My Matrix Calculation Correct for Composite Reflections in Linear Algebra?

[flyingpig]

Homework Statement

Find the standard matrix of T

$$T: \mathbb{R}^2 \to \mathbb{R}^2$$ first reflects points through the horizontal x₁ axis and then reflects points through the line x₂ = x₁

The Attempt at a Solution

I almost always have to look at my book to remind myself what the standard matrix transformation for each transformation is

So I did

e₁ becomes (which I will use $$\to$$) $$\to$$ e₁ (which doesn't change $$\to$$ e₂

That's column one. Also I am using David Lay's book, so I don't know words like "nullspace", "basis", etc...

$$e_2 \to -e_2 \to -(e_1)$$

Then I get

$$\begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix}$$

Now my question is, am I right about the $$-(e_1)$$? I added brackets on purpose

 

[Mark44]

Yes, that's right. Note that this matrix, with both transformations, is the product of two matrices:
$$\begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix}\begin{bmatrix}1 & 0\\ 0 & -1\end{bmatrix}$$
Let's call these matrices A and B, where A reflects across the line y = x (i.e., it switches the x and y coordinates) and B reflects across the x-axis.

The transformation that reflects across the x-axis and then reflects across the line y = x, can be expressed as the product of A and B, in that order. The expression ABv is evaluated as A(Bv), where Bv reflects v across the x-axis, and then A reflects the vector Bv across the line y = x.