Answer to Question #222316 in Linear Algebra for xiang

Question

Show that the dim R⁵=5

Answer 1

If U is a subspace of W then

D1: dim U ≤ dim W, and

D2: if dim U = dim W then U = W

Proof: Let $u_1.....u_k$ be a basis for U.

By Superset-Basis Lemma, there is a basis B for W that contains $u_1.....u_k$

$B = \{u_1...u_k;b_1...b_r\} \\ Thus \space k ≤ |B|, and\\ If \space k = |B| \space then \{u_1...u_k\} = B$

This implies that R^B=B

So, R⁵=5