Nullspace

Let A be a mxn matrix. The nullspace is therefore the set of vectors $\vec{x}$ such that $A\vec{x} = \vec{0}$

Nul(A)

A = $\begin{bmatrix} 1&0&1\\2&-1&1\end{bmatrix}$

Is $b = \begin{bmatrix} 2\\-2 \end{bmatrix} \in$ Nul(A)

NO! $A*b \neq \vec{0}$. Actually, we cannot even perform the operation.

It can therefore be seen that **THE NULLSPACE LIVES IN $\Re^n$**

But can we describe a null set?

Sure! We've been doing this a while without knowing it.

If we augment a matrix with the zero vector and solve down as we've been doing for some time, we find a set of vectors which describe the nullspace of the matrix A

Columnspace

The set of all linear combinations of the columns of A

A = $\begin{bmatrix} 1&0&1\\2&-1&1 \end{bmatrix}$

Therefore,

Col(A) = span{$\begin{bmatrix} 1\\2 \end{bmatrix} \begin{bmatrix} 0\\-1 \end{bmatrix} \begin{bmatrix} 1\\1 \end{bmatrix}$}

In general, we can therefore say the column space lives in $\Re^m$