# Thread: Span of Solution Set

1. ## Span of Solution Set

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!!!

2. Originally Posted by divinelogos
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!!!
Can you try asking the question again, maybe more completely how it's worded in the book, or wherever it came from? I'm not really understanding it...

3. 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

4. Originally Posted by divinelogos
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!!!
Now just decompose this vector into its parts

$\begin{pmatrix} -6z+w \\ y \\ z \\ w \end{pmatrix}=z\begin{pmatrix} -6\\ 0 \\ 1 \\ 0\end{pmatrix}+w\begin{pmatrix} 1 \\ 0 \\ 0 \\ 1\end{pmatrix}+y\begin{pmatrix} 0 \\ 1 \\ 0 \\ 0\end{pmatrix}$

So these 3 vectors span the nullspace