###### Exercise10

In this section, we saw that the solution space to the homogeneous equation $$A\xvec = \zerovec$$ is a subspace of $$\real^p$$ for some $$p\text{.}$$ In this exercise, we will investigate whether the solution space to another equation $$A\xvec = \bvec$$ can form a subspace.

Let's consider the matrix

\begin{equation*} A = \left[\begin{array}{rr} 2 \amp -4 \\ -1 \amp 2 \\ \end{array}\right]\text{.} \end{equation*}
1. Find a parametric description of the solution space to the homogeneous equation $$A\xvec = \zerovec\text{.}$$

2. Graph the solution space to the homogeneous equation to the right.

3. Find a parametric description of the solution space to the equation $$A\xvec = \twovec{4}{-2}$$ and graph it above.

4. Is the solution space to the equation $$A\xvec = \twovec{4}{-2}$$ a subspace of $$\real^2\text{?}$$

5. Find a parametric description of the solution space to the equation $$A\xvec=\twovec{-8}{4}$$ and graph it above.

6. What can you say about all the solution spaces to equations of the form $$A\xvec = \bvec$$ when $$\bvec$$ is a vector in $$\col(A)\text{?}$$

7. Suppose that the solution space to the equation $$A\xvec = \bvec$$ forms a subspace. Explain why it must be true that $$\bvec = \zerovec\text{.}$$

in-context