<s>[INST] Finding the conditions of (x,y,z,t) for them to belong to the span of a set of vectors So I got this math exercise, and I don't know how to go about it:
In $\mathbb{R}^4$, $S$ is the subspace spanned by the following set of vectors:  $(1, 1, 1, 0) , (1, 2, 1, 1) , (2, 0, 1, 1) , (3, 0, 4, 2)$
Find the condition (or a series of conditions) that $(x, y, z, t)$ must meet to so they belong to $S$.
 [/INST] If a vector is in the subspace spanned by those four vectors, then we have:
$$\begin{pmatrix} x\\ y\\ z\\ t\\
\end{pmatrix} =\alpha \begin{pmatrix}1\\1\\1\\0\\ \end{pmatrix}+\beta \begin{pmatrix}1\\2\\1\\1\\ \end{pmatrix} + \gamma \begin{pmatrix}2\\0\\1\\1\\\end{pmatrix} + \delta \begin{pmatrix}3\\0\\4\\2\\\end{pmatrix}$$ 
For some coefficients $\alpha, \beta, \gamma, \delta$. This is just saying that it can be expressed as a linear combination of these vectors. 
In the context of linear algebra, this can be expressed as 
$$ \begin{pmatrix}1&1&2&3\\1&2&0&0\\1&1&1&4\\0&1&1&2\\ \end{pmatrix} \begin{pmatrix} \alpha\\ \beta\\ \gamma\\ \delta\\
\end{pmatrix}=\begin{pmatrix} x\\ y\\ z\\ t\\
\end{pmatrix} $$
This should allow you to find your answer.
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