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.....uk be a basis for U.
By Superset-Basis Lemma, there is a basis B for W that contains u1.....uk
B={u1...uk;b1...br}Thus k≤∣B∣,andIf k=∣B∣ then{u1...uk}=B
This implies that RB=B
So, R5=5
Comments