###### Exercise 10

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

Find a parametric description of the solution space to the homogeneous equation \(A\xvec = \zerovec\text{.}\)

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

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

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

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

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

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