Standard Matrix of T: Horizontal Shear and Reflection Transformation Explained

[flyingpig]

Homework Statement

Find the standard matrix of T

$$T: \mathbb{R}^2 \to \mathbb{R}^2$$ first performs a horizontal shear tha transforms e₂ into e₂ - 2e₁ (leaving e₁) unchanged) and then reflects points through the line x₂ = x₁

The Attempt at a Solution

$$e_1 = \begin{bmatrix}<br /> 1\\ <br /> 0<br /> \end{bmatrix}$$

$$e_2 = \begin{bmatrix}<br /> 0\\<br /> 1<br /> \end{bmatrix}$$

$$e_2 - 2e_1 = \begin{bmatrix}<br /> 2\\<br /> 1<br /> \end{bmatrix}$$

$$A= \begin{bmatrix}<br /> -2 & 1 \\ <br /> 1 & 0<br /> \end{bmatrix}$$

Now I am completely stuck, how do I do the reflection? I know the standard matrix is just

A= $$\begin{bmatrix}<br /> 0 & -1 \\ <br /> -1 & 0<br /> \end{bmatrix}$$

But how do I "add" this information to my old standard matrix?

 

[LCKurtz]

Unless your notation is somehow different than mine, your A doesn't look correct.

Does Ae₁ = e₁?

And the answer to your other question is that if the first transform is accomplished by matrix A and the next by matrix B, then you want to calculate the matrix BA to get the matrix that does it in one step.

 

[HallsofIvy]

The "reflection through the line $x_2= x_1$" maps (1, 0) to (0, 1) and (0, 1) to (1, 0) so your matrix is
$$B= \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$$

I have called that matrix "B" because you have already use "A" for the "shear". The composition of the two transformations is the matrix product BA.