# Dimensions of complementary subspaces of V

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

## 2 Answers

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.

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