Question #222316

Show that the dim R5=5


1
Expert's answer
2021-08-17T10:15:06-0400

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 u1.....uku_1.....u_k be a basis for U.

By Superset-Basis Lemma, there is a basis B for W that contains u1.....uku_1.....u_k

B={u1...uk;b1...br}Thus kB,andIf k=B then{u1...uk}=BB = \{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 RB=B

So, R5=5


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