<s>[INST] Finding all values such that column vector is a linear combination Question:
For which values(s) of $ \ a$ is the column $ \ c = \begin{bmatrix} a\\a^2\\0\\a+1\end{bmatrix}$ a linear combination of the columns of,
$ \ x = \begin{bmatrix} 1\\1\\1\\1\end{bmatrix}$, $\ y = \begin{bmatrix} 1\\0\\-1\\0\end{bmatrix}$, $ \ z= \begin{bmatrix} 0\\1\\0\\1\end{bmatrix}$.
My attempt:
$ \ c = \begin{bmatrix} a\\a^2\\0\\a+1\end{bmatrix} = \ r\begin{bmatrix} 1\\1\\1\\1\end{bmatrix} + s \begin{bmatrix} 1\\0\\-1\\0\end{bmatrix} + t \begin{bmatrix} 0\\1\\0\\1\end{bmatrix} $
We know its a linear combination $\iff $   $ \ r+s = a, r+t = a^2, r-s = 0, r+t = a+1$  . I am not sure how to find all values of $ \ a$ now. 
 [/INST] $\require{cancel}$
Using your four obtained equations, you know that $r=s$ and so $2r=a.$ You then have $t=a^2-\frac{a}{2}.$ Inserting all that into the last equation, you get 
$$\cancel{\frac{a}{2}}+a^2-\cancel{\frac{a}{2}}=a+1,$$
the solutions of which are $\frac{1\pm\sqrt{5}}{2}.$
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