Problem:
A= [1 0 6 -1 0] has the solution set (this is a column vector) [6z+w y z w]
how do you find the span of this solution set? I already have the answer but am not sure how to get it.
Thanks for any help!!!
Sure!
The problem is:
Determine the null space of A and verify the Rank Nullity Theorem.
A=[1 0 -6 -1]
The null space is the solution set to the homogeneous system given by adding the 0:
A=[1 0 -6 -1 0]
there's 3 free variables y,z,w so the solution set is
x=-0y + 6z + w
To verify the Rank Nullity Theorem we have to get the nullity which involves finding the span, and hence the dimension (nullity).
My question is how do you get the span from that solution set?
See the attached picture
Now just decompose this vector into its parts
$\displaystyle \begin{pmatrix} -6z+w \\ y \\ z \\ w \end{pmatrix}=z\begin{pmatrix} -6\\ 0 \\ 1 \\ 0\end{pmatrix}+w\begin{pmatrix} 1 \\ 0 \\ 0 \\ 1\end{pmatrix}+y\begin{pmatrix} 0 \\ 1 \\ 0 \\ 0\end{pmatrix}$
So these 3 vectors span the nullspace