## Prove a Given Subset is a Subspace and Find a Basis and Dimension

## Problem 270

Let

\[A=\begin{bmatrix}

4 & 1\\

3& 2

\end{bmatrix}\]
and consider the following subset $V$ of the 2-dimensional vector space $\R^2$.

\[V=\{\mathbf{x}\in \R^2 \mid A\mathbf{x}=5\mathbf{x}\}.\]

**(a)** Prove that the subset $V$ is a subspace of $\R^2$.

**(b)** Find a basis for $V$ and determine the dimension of $V$.