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 &amp; 1 \\ <br /> 1 &amp; 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 &amp; -1 \\ <br /> -1 &amp; 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 &amp; 1 \\ 1 &amp; 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.