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Consider

[itex]

\[\left(\begin{array}{c}

0\\

0\\

0\end{array}\right)\] ,

\[\left(\begin{array}{c}

1\\

1\\

0\end{array}\right)\] ,

\[\left(\begin{array}{c}

1\\

0\\

1\end{array}\right)\] ,

\[\left(\begin{array}{c}

0\\

1\\

1\end{array}\right)\] as a subspace of \mathbb{Z}_{2}^{3}

[/itex]

In my notes I've written that this is a 2 dimensional subspace. How?

As far as I can see they are all linealry dependent vectors as if you add 1 of each of them you get back to the zero vector. No?