Question: 1. Let {x 1 , x 2 , ... ,x n } be any set of n vectors in an n-dimensional vector space V. Prove

1.

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

Prove that V=span{x1, x2, ... ,xn} if and only if{x1, x2, ... ,xn} 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)

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