1. ## size of null(A)

Hi all,
I've been asked a question in an assignment that I find a bit vague. In it we are given a matrix:

$\displaystyle \begin{bmatrix}1 & 1 & 0 & 1 & 0\\-1 & 1 & 1 & 2 & 3\\1 & 1 & 0 & 3 & 2\\-1 & 1 & 1 & 4 & 5\end{bmatrix}$

And it asks for the size of the nullspace and columnspace. I can find the nullspace and columnspace quite easily, but I am unsure what they want in regard to size.

the nullspace is:

$\displaystyle x_3\begin{bmatrix}0.5 \\ -0.5 \\ 1 \\ 0\end{bmatrix} + x_5\begin{bmatrix}1 \\ 0 \\ 0 \\ -1\end{bmatrix}$

the columnspace:

$\displaystyle \begin{bmatrix}1\\-1\\1\\-1\end{bmatrix}, \begin{bmatrix}1\\1\\1\\1\end{bmatrix}, \begin{bmatrix}1\\2\\3\\4\end{bmatrix}$

my initial reaction is that the nullspace is size of 2 (2 vectors create it), subsequently the columnspace is size of 3 (3 vectors create it). Can anyone confirm/deny+correct this?

2. Originally Posted by isp_of_doom
Hi all,
I've been asked a question in an assignment that I find a bit vague. In it we are given a matrix:

$\displaystyle \begin{bmatrix}1 & 1 & 0 & 1 & 0\\-1 & 1 & 1 & 2 & 3\\1 & 1 & 0 & 3 & 2\\-1 & 1 & 1 & 4 & 5\end{bmatrix}$

And it asks for the size of the nullspace and columnspace. I can find the nullspace and columnspace quite easily, but I am unsure what they want in regard to size.

the nullspace is:

$\displaystyle x_3\begin{bmatrix}0.5 \\ -0.5 \\ 1 \\ 0\end{bmatrix} + x_5\begin{bmatrix}1 \\ 0 \\ 0 \\ -1\end{bmatrix}$

the columnspace:

$\displaystyle \begin{bmatrix}1\\-1\\1\\-1\end{bmatrix}, \begin{bmatrix}1\\1\\1\\1\end{bmatrix}, \begin{bmatrix}1\\2\\3\\4\end{bmatrix}$

my initial reaction is that the nullspace is size of 2 (2 vectors create it), subsequently the columnspace is size of 3 (3 vectors create it). Can anyone confirm/deny+correct this?

Perhaps they meant "size" = dimension...

Tonio