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## Homework Statement

Prove that if W is a subspace of a finite-dimensional space V, then (dimension of W) is less than or equal to (dimension of V).

## The Attempt at a Solution

I've been trying to prove this by contradiction (assume dimension of W is greater than V and arrive at a contradiction) but am a little unclear on how to write the proof out. Here is my attempt so far:

W is a subspace of V. Assume S is a basis for W. If span(S) = V then dim(W) = dim(V).

Otherwise lets assume that dim(W) > dim(V).

Let S = {V1,V2,...,Vn} be a basis for W. Let T = {V1,V2,...,Vk) be a basis for V where, because of our assumption, n>k (there are more vectors in the basis S than in T).

Here is where i'm not sure where to proceed. I'm trying to see where i can use the fact that S is a subset of V and compare the number of vectors in S and T but i'm just not seeing the answer. Please help me understand the next step conceptually. Thank you for your time.