1.

Let {x₁, x₂, ... ,x_n} be any set of n vectors in an n-dimensional vector space V.

Prove that V=span{x₁, x₂, ... ,x_n} if and only if{x₁, x₂, ... ,x_n} is linearly independent.

2.

Let V be a vector space of dimension n. Suppose that W₁ and W₂ be subspaces of V, W₁W₂, and that dim(W₁)=dim(W₂)=n-1.

Prove that V=W₁+W₂ and determine dim(W₁W₂)