Question

Exercise 2.20 Let S= {v, V2, ... , Vpf be a subset of R" containing n vectors. Prove that if S generates R and S is linearly independent, then n m.

Answer 1

Proof:

Let  $\small B$ be the basis of $\small \mathbb{R}^{m}$ with  $\small m$ elements.

Since  $\small S$ generates  $\small \mathbb{R}^{m}$ and  $\small S$ is linearly independent,  $\small S$ is also a basis of  $\small \mathbb{R}^{m}$ .

If
n m
then $\small S$ is linearly dependent.

Hence   .

n m

If  
mn
then $\small B$ is linearly dependent.

Hence   .

m <n

Thus,   .

m