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I'm trying to show that for U1 and U2 which are subsets of a vector space V

If U1 is p-dimensional subspace of V, and V is n-dimensional, then there exists a complementary subspace U2 which has dimension n-p.

I've started by saying that the basis of U1 is a set of p linearly independent vectors in V, and the basis of U2 is a set of m linearly independent vectors in V.

But I don't think I really understand how to use the definition of the complementary subspace (U1 + U2 = V , intersection of U1 and U2 = {0}) to go any further

Thanks

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You can start with a basis of $U$, and add in suitable vectors one by one until you have a basis of $V$. These additional vectors will form a basis of a complementary subspace.

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I hope that I understand your question

Choose all basis vectors in $V$ such that they are not in U_1. Define $U_2 $like linear span of these vectors.

$U_1 \cap U_2 = 0$ because their basis are linear independent

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