Activity 3.5.3

We will explore some column spaces in this activity.

1. Consider the matrix $$A= \left[\begin{array}{rrr} \vvec_1 & \vvec_2 & \vvec_3 \end{array}\right] = \left[\begin{array}{rrr} 1 & 3 & -1 \\ -2 & 0 & -4 \\ 1 & 2 & 0 \\ \end{array}\right].$$ Since $\col(A)$ is the span of the columns, we have $$\col(A) = \laspan{\vvec_1,\vvec_2,\vvec_3}.$$ Explain why $\vvec_3$ can be written as a linear combination of $\vvec_1$ and $\vvec_2$ and why $\col(A)=\laspan{\vvec_1,\vvec_2}$.

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

3. Now consider the matrix $B$ and its reduced row echelon form: $$B = \left[\begin{array}{rrrr} -2 & -4 & 0 & 6 \\ 1 & 2 & 0 & -3 \\ \end{array}\right] \sim \left[\begin{array}{rrrr} 1 & 2 & 0 & -3 \\ 0 & 0 & 0 & 0 \\ \end{array}\right].$$ Explain why $\col(B)$ is a 1-dimensional subspace of $\real^2$ and is therefore a line.

4. For a general matrix $A$, what is the relationship between the dimension $\dim~\col(A)$ and the number of pivot positions in $A$?

5. How does the location of the pivot positions indicate a basis for $\col(A)$?

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

7. Suppose that $A$ is an $8\times 10$ matrix and that $\col(A) = \real^8$. If $\bvec$ is an 8-dimensional vector, what can you say about the equation $A\xvec = \bvec$?