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Prove or disprove:

If U is a subspace of a finite dimensional vector space V and B = {v1, . . . ,vn} is a basis for V, then some subset of B is a basis for U.

So far, I don't know where to start. I could assume that since B is a basis, it is linearly independent, and thus, some subset of B, containing less vectors than B, could potentially be a basis for U, since U is a subspace of V, but I don't know how to go about proving this.

Thanks

Ian Murphy
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2 Answers2

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It's not true. Suppose $V=\mathbb R^2$ Let the basis be $(0,1),(1,0)$ and the subspace be generated by $(1,1)$

Gregory Grant
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Hint: if this statement is generally true, then it is true in the case $U$ is a one-dimensional subspace, i.e. it is just the span of a single vector.

If a subset of $B$ is then a basis for $U$, what does this tell you about the vectors in $B$?

BaronVT
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