Activity 3.5.4
We will explore some column spaces in this activity.

Consider the matrix
\begin{equation*} A= \left[\begin{array}{rrr} \vvec_1 \amp \vvec_2 \amp \vvec_3 \end{array}\right] = \left[\begin{array}{rrr} 1 \amp 3 \amp 1 \\ 2 \amp 0 \amp 4 \\ 1 \amp 2 \amp 0 \\ \end{array}\right]\text{.} \end{equation*}Since \(\col(A)\) is the span of the columns, the vectors \(\vvec_1\text{,}\) \(\vvec_2\text{,}\) and \(\vvec_3\) naturally span \(\col(A)\text{.}\) Are these vectors linearly independent?
Show that \(\vvec_3\) can be written as a linear combination of \(\vvec_1\) and \(\vvec_2\text{.}\) Then explain why \(\col(A)=\span{\vvec_1,\vvec_2}\text{.}\)
Explain why the vectors \(\vvec_1\) and \(\vvec_2\) form a basis for \(\col(A)\text{.}\) This shows that \(\col(A)\) is a 2dimensional subspace of \(\real^2\) and is therefore a plane.

Now consider the matrix \(A\) and its reduced row echelon form:
\begin{equation*} A = \left[\begin{array}{rrrr} 2 \amp 4 \amp 0 \amp 6 \\ 1 \amp 2 \amp 0 \amp 3 \\ \end{array}\right] \sim \left[\begin{array}{rrrr} 1 \amp 2 \amp 0 \amp 3 \\ 0 \amp 0 \amp 0 \amp 0 \\ \end{array}\right]\text{.} \end{equation*}We will call the columns \(\vvec_1\text{,}\) \(\vvec_2\text{,}\) \(\vvec_3\text{,}\) and \(\vvec_4\text{.}\) Explain why \(\vvec_2\text{,}\) \(\vvec_3\text{,}\) and \(\vvec_4\) can be written as a linear combination of \(\vvec_1\text{.}\)
Explain why \(\col(A)\) is a 1dimensional subspace of \(\real^2\) and is therefore a line.
What is the relationship between the dimension \(\dim~\col(A)\) and the rank \(\rank(A)\text{?}\)
What is the relationship between the dimension of the column space \(\col(A)\) and the null space \(\nul(A)\text{?}\)
If \(A\) is an invertible \(9\times9\) matrix, what can you say about the column space \(\col(A)\text{?}\)
If \(\col(A)=\{\zerovec\}\text{,}\) what can you say about the matrix \(A\text{?}\)