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I'm having difficulty with this question:

Let V be a subspace of $\mathbb R^4$ spanned by the set $U = {(1, -1, 3, 1), (2, 1, -1, 2), (-1, 3, 0, 2)}$. Show that U is a basis of V and determine whether the vector $t = (-3, 6, 7, 6)$ belongs to space V in order to find the coordinate vector of $t$ relative to basis $U$.

My problem is that although the vectors are linearly independent with the trivial solution $c$1 = $c$2 = $c$3 = $0$, I did not think it was possible to span $\mathbb R^4$ with only 3 vectors.

I assume $t$ can be found by equating the vectors in $U$ to $t$ for the $b$ column of the matrix, but how can a coordinate vector be found if it seems that $U$ is not a basis of $V$? Or have I made a wrong assumption?

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    $\begingroup$ You are correct that it is not possible to span $\mathbb R^4$ with only $3$ vectors, but the question stipulates that $V$ is a subspace of $\mathbb R^4$ spanned by the $3$ vectors in $U$ $\endgroup$ Aug 13 '20 at 5:36
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You are correct that it is not possible to span $\mathbb R^4 $ with only $3$ vectors,

but the question stipulates that $V$ is a subspace of $\mathbb R^4$ spanned by the $3$ vectors in $U$.

In fact, $t=2(1,-1,3,1)-(2,1,-1,2)+3(-1,3,0,2)$ is in $V$, though not every vector in $\mathbb R^4$ is.

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  • $\begingroup$ So, I only need to show linear independence to show it is a basis then? And then find the coordinate vector from that? $\endgroup$
    – DuncanK3
    Aug 14 '20 at 8:26

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