###### Activity3.5.3

We will explore some null spaces in this activity.

1. Consider the matrix

\begin{equation*} A=\left[\begin{array}{rrr} 1 \amp 3 \amp -1 \\ -2 \amp 0 \amp -4 \\ 1 \amp 2 \amp 0 \\ \end{array}\right] \end{equation*}

and give a parametric description of the null space $$\nul(A)\text{.}$$

2. Give a basis for and state the dimension of $$\nul(A)\text{.}$$

3. The null space $$\nul(A)$$ is a subspace of $$\real^p$$ for which $$p\text{?}$$

4. Now consider the matrix $$A$$ whose reduced row echelon form is given:

\begin{equation*} A \sim \left[\begin{array}{rrrr} 1 \amp 2 \amp 0 \amp -3 \\ 0 \amp 0 \amp 1 \amp 2 \\ \end{array}\right]\text{.} \end{equation*}

Give a parametric description of $$\nul(A)\text{.}$$

5. Notice that the parametric description gives a set of vectors that span $$\nul(A)\text{.}$$ Explain why this set of vectors is linearly independent and hence forms a basis. What is the dimension of $$\nul(A)\text{?}$$

6. For this matrix, $$\nul(A)$$ is a subspace of $$\real^p$$ for what $$p\text{?}$$

7. What is the relationship between the dimensions of the matrix $$A\text{,}$$ the number of pivot positions of $$A$$ and the dimension of $$\nul(A)\text{?}$$

8. Suppose that the columns of a matrix $$A$$ are linearly independent. What can you say about $$\nul(A)\text{?}$$

9. If $$A$$ is an invertible $$n\times n$$ matrix, what can you say about $$\nul(A)\text{?}$$

10. Suppose that $$A$$ is a $$5\times 10$$ matrix and that $$\nul(A) = \real^{10}\text{.}$$ What can you say about the matrix $$A\text{?}$$

in-context