Please explain the proof and importance of Theorem 4.5.2

Question

 Please explain the proof and importance of Theorem 4.5.2: Let V be a finite-dimensional vector space, and let {v₁, v₂, ... v_n} be any basis for V.

(a) If a set has more than n vectors, then it is linearly dependent.

(b) If a set has fewer than n vectors, then it does not span V.

Answer 1

Here are some informal explanations that show why the statements are true.

1.

First of all, we know that if v1,v2,...,vn are a basis for V, then for any nonzero v in V, there exists unique constants (not all zero) such that

c1v1+c2v2+...+cnvn = v. Immediately, this shows that v depends linearly on v1,v2,...,vn.

So, if I have a set {v1,v2,...,vn,v}, I know immediately that v depends on v1,v2,...,vn (and thus all the vectors in the set are linearly independent).

2.

Suppose that I have n vectors in my basis: v1,v2,...,vn. By definition, these vectors are linearly independent. In particular, vn cannot be written as a linear combination of v1,...,v(n-1). Thus, if my set is {v1,v2,...,v(n-1)}, then vn is in V, but outside the range of my set. Therefore, my set does not span V.