<s>[INST] Question about the basis of an intersection of subspaces 
Let $ U=\left\langle\left(\begin{array}{c}-5 \\ 2 \\ 0\end{array}\right),\left(\begin{array}{c}-2 \\ 2 \\ -1\end{array}\right)\right\rangle $ and $ W=\left\langle\left(\begin{array}{c}-3 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{c}-1 \\ -2 \\ -2\end{array}\right)\right\rangle 
$ be vector subspaces of $ \mathbb{R}^{3} $.
a) Determine the basis and dimension of $U, W$ and $U\cap W$.
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Solution:
$ B_{U}=\left\{\left(\begin{array}{c}-5 \\ 2 \\ 0\end{array}\right),\left(\begin{array}{c}-2 \\ 2 \\ -1\end{array}\right)\right\} \implies \dim(U)=2$
$ B_{W}=\left\{\left(\begin{array}{c}-3 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{c}-1 \\ -2 \\ -2\end{array}\right)\right\} \implies \dim(W)=2$
$ B_{U\cap W}=\left\{\left(\begin{array}{c}-3 \\ 0 \\ 1\end{array}\right)\right\} \implies \dim(U\cap W)=1$
My question: how do I get to $B_{U\cap W}$? I do not understand the solution. Could someone please explain?
 [/INST] Hint :
You write the equation :
$$\lambda_{1}\begin{pmatrix}-5 \\ 2 \\ 0\end{pmatrix}+\lambda_{2}\begin{pmatrix}-2 \\ 2 \\ -1\end{pmatrix}
= \gamma_{1}\begin{pmatrix}-3 \\ 0 \\ 1\end{pmatrix} +\gamma_{2}\begin{pmatrix}-1 \\ -2 \\ -2\end{pmatrix}$$
And after you get a very simple linear system 
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