Activity 3.5.3
We will explore some null spaces in this activity.

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 \(\nul(A)\text{.}\)
Give a basis for and state the dimension of \(\nul(A)\text{.}\)
The null space \(\nul(A)\) is a subspace of \(\real^p\) for which \(p\text{?}\)

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 \(\nul(A)\text{.}\)
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 \(\nul(A)\text{?}\)
For this matrix, \(\nul(A)\) is a subspace of \(\real^p\) for what \(p\text{?}\)
What is the relationship between the dimensions of the matrix \(A\text{,}\) the number of pivot positions of \(A\) and the dimension of \(\nul(A)\text{?}\)
Suppose that the columns of a matrix \(A\) are linearly independent. What can you say about \(\nul(A)\text{?}\)
If \(A\) is an invertible \(n\times n\) matrix, what can you say about \(\nul(A)\text{?}\)
Suppose that \(A\) is a \(5\times 10\) matrix and that \(\nul(A) = \real^{10}\text{.}\) What can you say about the matrix \(A\text{?}\)