Let V be a finite dimensional vector space Suppose U is a su

Let V be a finite dimensional vector space. Suppose U is a subspace of V and dim U = dim V. Prove that U = V.

Solution

Solution : 9)

Let { U₁,...,U_m }  is a basis for U  and dimV = n since dimU = dimV, m = n. We have, any linearly independent set with same number of vectors as a basis is a another basis. So { U₁,...,U_n } is a basis for V too. As a subspace, U V So only to show that V U. Let vV. Then there are scalars ₁, ₂,...,_n such that v = ₁u₁ + ... + _nu_n as { U₁,...,U_m } is a basis for U, vU, so V U. Which shows U = V