###### Activity3.5.4

We will explore some column spaces in this activity.

1. 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?

2. Show that $$\vvec_3$$ can be written as a linear combination of $$\vvec_1$$ and $$\vvec_2\text{.}$$ Then explain why $$\col(A)=\span{\vvec_1,\vvec_2}\text{.}$$

3. Explain why the vectors $$\vvec_1$$ and $$\vvec_2$$ form a basis for $$\col(A)\text{.}$$ This shows that $$\col(A)$$ is a 2-dimensional subspace of $$\real^2$$ and is therefore a plane.

4. 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 $$\vvec_1\text{.}$$

5. Explain why $$\col(A)$$ is a 1-dimensional subspace of $$\real^2$$ and is therefore a line.

6. What is the relationship between the dimension $$\dim~\col(A)$$ and the rank $$\rank(A)\text{?}$$

7. What is the relationship between the dimension of the column space $$\col(A)$$ and the null space $$\nul(A)\text{?}$$

8. If $$A$$ is an invertible $$9\times9$$ matrix, what can you say about the column space $$\col(A)\text{?}$$

9. If $$\col(A)=\{\zerovec\}\text{,}$$ what can you say about the matrix $$A\text{?}$$

in-context