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.

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  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 \(\bvec = \zerovec\text{.}\)

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