I have this problem that I am doing in a book, and well, the answer in the back says the dimension of the column space and the row space is two.

$\displaystyle \left(

\begin{matrix}

1 & 1 & 0\\

1 & 3 & 1\\

3 & 1 & -1\\

\end{matrix}

\right)

$

I row reduced it to:

$\displaystyle \left(

\begin{matrix}

1 & 0 & 0\\

0 & 1 & 0\\

0 & 0 & 1\\

\end{matrix}

\right)

$

So I thought this would mean that they are independent columns and the space it spans has a dimension of 3.

Am I missing something here?