We are given the transformation $T(x, y, z)$ as follows:

$T(x, y, z) = \begin{bmatrix} x + z \\ 2y \end{bmatrix}$

Problem 1: Find a matrix $A$ such that $T(x) = A x$.

To represent this transformation as a matrix, we will express $T(x, y, z)$ as a matrix multiplication. Assume $x$, $y$, and $z$ are represented as a column vector:

$\begin{bmatrix} x \\ y \\ z \end{bmatrix}$

We are looking for a matrix $A$ such that:

$T(x, y, z) = A \begin{bmatrix} x \\ y \\ z \end{bmatrix}$

Step-by-Step Solution:

Step 1: Write the transformation in terms of components

The transformation is given as:

$T(x, y, z) = \begin{bmatrix} x + z \\ 2y \end{bmatrix}$

We want to find a matrix $A$ that transforms the vector $\begin{bmatrix} x \\ y \\ z \end{bmatrix}$ into $\begin{bmatrix} x + z \\ 2y \end{bmatrix}$.

Step 2: Analyze the first component

The first component of the output vector is $x + z$. This suggests that the first row of matrix $A$ should take the $x$-coordinate and add the $z$-coordinate, while ignoring the $y$-coordinate. Therefore, the first row of $A$ will be:

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

This row represents the transformation of the input vector to produce the first component $x + z$.

Step 3: Analyze the second component

The second component of the output vector is $2y$. This suggests that the second row of matrix $A$ should take the $y$-coordinate, multiply it by 2, and ignore the $x$- and $z$-coordinates. Therefore, the second row of $A$ will be:

$\begin{bmatrix} 0 & 2 & 0 \end{bmatrix}$

This row represents the transformation of the input vector to produce the second component $2y$.

Step 4: Construct the matrix $A$

Now, we combine the two rows we derived into a single matrix. The matrix $A$ is:

$A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 2 & 0 \end{bmatrix}$

Thus, the transformation $T(x, y, z)$ can be written as:

$T(x, y, z) = A \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 2 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix}$

Verifying:

$A \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \cdot x + 0 \cdot y + 1 \cdot z \\ 0 \cdot x + 2 \cdot y + 0 \cdot z \end{bmatrix} = \begin{bmatrix} x + z \\ 2y \end{bmatrix}$

This confirms that the matrix $A$ is correct.

Would you like more details or have any further questions? Here are 5 related questions to consider:

1. How would the matrix change if the transformation was $T(x, y, z) = \begin{bmatrix} x + 2z \\ 3y \end{bmatrix}$?
2. How do you find the inverse of matrix $A$, assuming it exists?
3. What are the conditions for a matrix transformation to be invertible?
4. How can you interpret matrix transformations geometrically in 3D space?
5. How do you compute the determinant of matrix $A$, and what does it represent?

Tip: Understanding matrix transformations helps in visualizing how vectors in space are stretched, rotated, or sheared through multiplication.