Oswaldo Riley

2023-03-31

I need to find a unique description of Nul A, namely by listing the vectors that measure the null space
$A=\left[\begin{array}{ccccc}1& 5& -4& -3& 1\\ 0& 1& -2& 1& 0\\ 0& 0& 0& 0& 0\end{array}\right]$

?

Chana Garner

To find the null space of the matrix $A=\left[\begin{array}{ccccc}1& -5& -4& -3& 1\\ 0& 1& -2& 1& 0\\ 0& 0& 0& 0& 0\end{array}\right]$, we need to solve the equation $A𝐱=\mathbf{0}$, where $𝐱$ is a vector in the null space of $A$.
Setting up the augmented matrix and performing row reduction, we have:
$\left[\begin{array}{ccccccc}1& -5& -4& -3& 1& |& 0\\ 0& 1& -2& 1& 0& |& 0\\ 0& 0& 0& 0& 0& |& 0\end{array}\right]$
Performing row operations, we obtain:
$\left[\begin{array}{ccccccc}1& 0& -6& 2& 1& |& 0\\ 0& 1& -2& 1& 0& |& 0\\ 0& 0& 0& 0& 0& |& 0\end{array}\right]$
From the row-reduced form, we can see that ${x}_{3}$ and ${x}_{5}$ are free variables. We can express the null space vectors as:
$𝐱=\left[\begin{array}{c}6s-2t\\ 2s-t\\ s\\ t\\ s\end{array}\right]$
where $s$ and $t$ are parameters.
In order to describe the null space, we can write it as the span of the following vectors:

Therefore, the null space of matrix $A$ is described by the set of vectors:

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