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. 

A necessary condition is:
$$a^2 = r+t = a+1 \implies a = \frac{1\pm\sqrt{5}}{2}$$
Then you have $2r = a \implies r = s = \frac{1\pm\sqrt{5}}{4}$ so in that case the rows are indeed linearly dependent.
Thus, the rows are linearly dependent iff $a = \frac{1\pm\sqrt{5}}{2}$
