If Av1, Av2, ..., Avp are linearly independent, prove p ≤ 10

[sweetiepi]

Homework Statement

Let A be a 2000 x 10 matrix and v1, v2, ... , vp vectors in R10. Suppose that Av1, Av2, ... , Avp are linearly independent vectors in R2000.

a) Prove that p is < or = to 10
b) Prove that if p = 10, the columns of A are linearly independent

Homework Equations

Given above

The Attempt at a Solution

At first my line of thinking was that the products Av1, Av2 etc each had 10 unknowns and that these were somehow all related so that if there were more than 10 terms of Av1, Av2, ... , Avp then the linear combination would be linearly dependent. But I think at this point I'm just confusing myself, and it's difficult for me to picture a linear combination of a linear combination... So any help would be greatly appreciated!

 

[ystael]

If $$\{Av_1, \dots, Av_p\}$$ is linearly independent, can $$\{v_1, \dots, v_p\}$$ be linearly dependent?