I understand $\mathbf{e}_2 + 24\mathbf{e}_1$, but I do not understand $x_2 = -x_1$. What is $x_2$ and $-x_1$ referring to? This is the part that is preventing me from doing these types of questions.
Assume that $T$ is a linear transformation. Find the standard matrix of $T$.
$T:\mathbb{R}^2\to\mathbb{R}^2$, first performs a horizontal shear that transforms $\mathbf{e}_2$ into $\mathbf{e}_2+24\mathbf{e}_1$ (leaving $\mathbf{e}_1$ unchanged) and then reflects points through the line $x_2=-x_1$.
$$\begin{bmatrix}\mathbf{e}_1 & \mathbf{e}_2+24\mathbf{e}_1
\end{bmatrix}= \begin{bmatrix} 1 & 24 \\ 0 & 1 \\ \end{bmatrix}
$$ The reflection through the line $x_2=-x_1$ maps $\mathbf{e}_1$ into $-\mathbf{e}_2$. Notice that this reflection is a linear transformation.This means $\mathbf{e}_1$ maps into $-\mathbf{e}_2$ and $\mathbf{e}_2+24\mathbf{e}_1$ maps into $-\mathbf{e}_1-24\mathbf{e}_2$.
Write the column vectors for the transformation $\mathbf{e}_1=-\mathbf{e}_2$ and $\mathbf{e}_2=-\mathbf{e}_1-24\mathbf{e}_2$.
Please explain what this part means, why, and how to solve the problem. Also, please include a diagram of if possible.
Thank You.
Best Answer
$x_1$ and $x_2$ are the components of a vector $\mathbf{x}$ with respect to the standard basis. This means that: $$ \mathbf{x}=[x_1,x_2]^T=x_1\mathbf{e_1}+x_2\mathbf{e_2} $$
(Maybe that you prefer the notation $x_1 =x$ and $x_2=y$ so that the vector is $\mathbf{x}=[x,y]^T$, but it is the same).
So the line $x_2=-x_1$( or $y=-x$) is the bisector of the second and fourth quadrant and the reflection through this line is represented by the matrix: $$ T_2=\begin{bmatrix} 0&-1\\-1&0 \end{bmatrix} $$ as you can easily see reflecting $\mathbf{e_1}$ and $\mathbf{e_2}$.
So your transformation is $T=T_2T_1$ with $$ T_1=\begin{bmatrix} 1&24\\0&1 \end{bmatrix} $$ and we have: $$ T=\begin{bmatrix} 0&-1\\-1&0 \end{bmatrix}\begin{bmatrix} 1&24\\0&1 \end{bmatrix}= \begin{bmatrix} 0&-1\\-1&-24 \end{bmatrix} $$