# Coordinate vector from a Basis of a subspace

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?

• 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 $U$ Aug 13, 2020 at 5:36

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.