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So basically I have a problem with 2 subspaces given in the following spans $$U=\mathscr L\{(1,2,-1,3),(2,4,1,-2),(3,6,3,-7)\}$$$$V=\mathscr L\{(1,2,-4,11),(2,4,0,14)\}$$ And I am asked if it is true if U$\subseteq$V or V$\subseteq$U. I understand the concept of 2 subspace being equal or proving a subset being a subspace of $\mathbb R^n$ by checking the sum of the vectors and the multiplication by a constant but I don't believe to understand the procedure to check if one subspace is contained in another one.

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If the generators of $V$ are in $U$, then $V \subseteq U$.

EDIT: in this particular case, you can do it faster. Call $u_1, u_2, u_3$ the generators of $U$. Then $u_2 - 2u_1 = 2(u_3 - 3u_1)$, so both $U$ and $V$ have dimension $2$. You just have to check if they are equal or not.

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