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I would really appreciate some help with trying to find the Cartesian equation of a subspace (part b). Have attached my answers. Thank you very much in advanced![my workings out and the question]1

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There should be two parameters, since it is spanned by two linearly independent vectors: $$\begin{pmatrix}x\\y\\z\end{pmatrix}=s\begin{pmatrix}1\\2\\0\end{pmatrix}+t\begin{pmatrix}1\\0\\1\end{pmatrix}=\begin{pmatrix}s+t\\2s\\t\end{pmatrix}.$$ This gives you $s=y/2$, $t=z$. Plugging this into the equation for $x$, we get $x=y/2+z$.

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    Thank you ever so much! Fantastic! Just realised that I just wrote out the line that passes through those two points, rather than the 2-d subspace as you have shown! – 2017-01-08
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    I.e. the hyperplan in $\mathds{R}^3$ – 2017-01-08
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    @Andy: right, what you wrote is the line passing through the two points. – 2017-01-08