Let's analyze and address each part of this problem step-by-step.

Given:

• Subspaces $V$ and $W$ of $M_{22}(\mathbb{R})$: $V = \text{span} \left\{ \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \right\}$ $W = \text{span} \left\{ \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \right\}$
• A linear map $f: V \to W$ defined by $f\left( \begin{pmatrix} x & y \\ 0 & z \end{pmatrix} \right) = \begin{pmatrix} x + y & z \\ z & x + y \end{pmatrix}$ for any $\begin{pmatrix} x & y \\ 0 & z \end{pmatrix} \in M_{22}(\mathbb{R})$.

Problem Breakdown

1. Prove that $f$ is linear.

   To show that $f$ is linear, we need to check two properties:

   • Additivity: $f(A + B) = f(A) + f(B)$ for all $A, B \in V$.
   • Scalar Multiplication: $f(\alpha A) = \alpha f(A)$ for any scalar $\alpha$ and $A \in V$.

2. Find a basis for $V + W$ and $V \cap W$.

   • To find $V + W$, we look for the smallest set that spans all elements of both $V$ and $W$.
   • To find $V \cap W$, we identify elements common to both $V$ and $W$.

3. Find a basis for the kernel $\ker(f)$ and the image $\text{Im}(f)$.

   • $\ker(f)$ is the set of elements in $V$ that $f$ maps to the zero matrix in $W$.
   • $\text{Im}(f)$ is the span of all matrices of the form $f(A)$ for $A \in V$.

Let's proceed with each part in more detail.

Solution Outline

1. Proof of Linearity of $f$: We need to show that $f$ preserves addition and scalar multiplication. That is:

   • For $A = \begin{pmatrix} x & y \\ 0 & z \end{pmatrix}$ and $B = \begin{pmatrix} x' & y' \\ 0 & z' \end{pmatrix}$: $f(A + B) = f\left( \begin{pmatrix} x + x' & y + y' \\ 0 & z + z' \end{pmatrix} \right) = \begin{pmatrix} (x + x') + (y + y') & z + z' \\ z + z' & (x + x') + (y + y') \end{pmatrix}$ and separately $f(A) + f(B) = \begin{pmatrix} x + y & z \\ z & x + y \end{pmatrix} + \begin{pmatrix} x' + y' & z' \\ z' & x' + y' \end{pmatrix} = \begin{pmatrix} (x + x') + (y + y') & z + z' \\ z + z' & (x + x') + (y + y') \end{pmatrix}.$ Therefore, $f(A + B) = f(A) + f(B)$.

   • For scalar $\alpha$: $f(\alpha A) = f\left( \begin{pmatrix} \alpha x & \alpha y \\ 0 & \alpha z \end{pmatrix} \right) = \begin{pmatrix} \alpha (x + y) & \alpha z \\ \alpha z & \alpha (x + y) \end{pmatrix} = \alpha f(A).$

   Thus, $f$ is linear.

2. Finding a Basis for $V + W$ and $V \cap W$:

   • Compute spanning sets for $V + W$ and common elements in $V \cap W$.

3. Finding Bases for $\ker(f)$ and $\text{Im}(f)$:

   • Solve $f(A) = 0$ for elements in $\ker(f)$.
   • Determine the span of $f(A)$ for $A \in V$ to find $\text{Im}(f)$.

Would you like a detailed, step-by-step solution for each of these parts?