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 $S$.
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.