Let's break down the problem to find the standard matrix $A$ for the transformation $T: \mathbb{R}^3 \to \mathbb{R}^2$.

Problem Summary:

• The transformation $T$ involves two steps:
  1. Projection onto the $yz$-plane: This means that the $x$-component of any vector in $\mathbb{R}^3$ becomes zero, leaving only the $y$- and $z$-components.
  2. Reflection around the line $y = -z$: After projecting onto the $yz$-plane, the vector is reflected around the line $y = -z$.

Step 1: Projection onto the $yz$-plane

To project any vector $\mathbf{v} = (x, y, z)$ onto the $yz$-plane, we eliminate the $x$-component. The projection matrix for this operation would be:

$P_{\text{yz}} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

This keeps the $y$- and $z$-components unchanged and sets the $x$-component to zero.

Step 2: Reflection around the line $y = -z$

The reflection around the line $y = -z$ exchanges and negates the $y$- and $z$-components. The reflection matrix for this is:

$R = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & -1 & 0 \end{pmatrix}$

Final Transformation Matrix

To combine the two operations, we first apply the projection matrix $P_{\text{yz}}$ and then the reflection matrix $R$. The overall transformation matrix $A$ is obtained by multiplying the two matrices:

$A = R \cdot P_{\text{yz}} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & -1 & 0 \end{pmatrix} \cdot \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

Let's calculate the result:

$A = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & -1 & 0 \end{pmatrix}$

This is the standard matrix for the transformation $T$.

If you have any questions or want further clarification, feel free to ask!

Related Questions:

1. What is the projection matrix for projecting onto the $xy$-plane instead of the $yz$-plane?
2. How does the reflection matrix change if we reflect around the line $y = z$?
3. Can you compute the effect of applying this transformation to the vector $(1, 2, 3)$?
4. How would the matrix change if the projection were onto the $xz$-plane instead?
5. How can we generalize the reflection matrix to reflect about an arbitrary line in the $yz$-plane?

Tip:

When working with compositions of linear transformations, always remember to multiply the matrices in the order the transformations are applied.