How to go about proving this statement?
Suppose V is finite dimensional and U is a subspace of V . Prove that there exists a subspace W of V such that V = U + W and U ∩ W = {0}, where 0 is the additive identity of V .
Proof of existence of subspaces that sum up to the vector space
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linear-algebra
abstract-algebra
vector-spaces
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0I have verified with certain examples, which hold true, I am looking for a formal proof to this. – 2017-02-18
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1Start with a basis of $U$, then extend it to a basis of $V$. Let $W$ be the span of the basis elements not in $U$. – 2017-02-18
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0Related : [see here](http://math.stackexchange.com/questions/513125/there-is-a-subspace-w-of-v-such-that-v-u-oplus-w) – 2017-02-18
1 Answers
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Take a basis $B$ of $U$, and complete this basis $B\cup B_c$ into a basis of $V$
$W = \text{Span}(B_c)$ is a possible answer