Activity 3.5.4

We will explore some null spaces in this activity and see why $\nul(A)$ satisfies the definition of a subspace.

1. Consider the matrix $$A=\begin{bmatrix} 1 & 3 & -1 & 2 \\ -2 & 0 & -4 & 2 \\ 1 & 2 & 0 & 1 \end{bmatrix}$$ and give a parametric description of the solution space to the equation $A\xvec = \zerovec$. In other words, give a parametric description of $\nul(A)$.

2. This parametric description shows that the vectors satisfying the equation $A\xvec=\zerovec$ can be written as a linear combination of a set of vectors. In other words, this description shows why $\nul(A)$ is the span of a set of vectors and is therefore a subspace. Identify a set of vectors whose span is $\nul(A)$.

3. Use this set of vectors to find a basis for $\nul(A)$ and state the dimension of $\nul(A)$.

4. The null space $\nul(A)$ is a subspace of $\real^p$ for which value of $p$?

5. Now consider the matrix $B$ whose reduced row echelon form is given by $$B \sim \left[\begin{array}{rrrr} 1 & 2 & 0 & -3 \\ 0 & 0 & 0 & 0 \\ \end{array}\right].$$ Give a parametric description of $\nul(B)$.

6. The parametric description gives a set of vectors that span $\nul(B)$. Explain why this set of vectors is linearly independent and hence forms a basis. What is the dimension of $\nul(B)$?

7. For a general matrix $A$, how does the number of pivot positions indicate the dimension of $\nul(A)$?

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