Bases, Invertibility and Injectivity Query

[Ad123q]

Hi

There are two things that are confusing me a bit, and was wondering if anyone could explain them.

Firstly, if we let V1,V2,...,Vn be vectors in some field F^n and let P = {V1,V2,...,Vn},
then the following are equivalent:

(i) {V1,V2,...,Vn} is a basis for F^n ;
(ii) P is invertible.

Secondly, let T be a linear map. Then if T is invertible, then it is injective.

Thanks for the hep in advance!

 

[CompuChip]

I'm not quite sure what you mean by "P is invertible".
As I see it, P is just a set (with n vectors as its elements). What does it mean for a set to be invertible? Or is there some definition / notation that is not generally used here?

For the second one I can help you: to show that T is injective, you should have that whenever T(x) = T(y) (for arbitrary x and y in the domain), then x = y. This is almost trivial to prove: apply the inverse of T to both sides!

 

[Cantab Morgan]

I think Ad123q means that P is the matrix whose column vectors V1, V2, etc.

If P is invertible then the V1, V2, etc must be linearly independent. Conversely, it is also true that if the V1, V2, etc are linearly independent, then the matrix P must be invertible.

To get a feel for this, suppose two columns of a matrix were linearly dependent. (Say the ith and jth.) Then that would mean that two linearly independent vectors would be mapped by P onto the same line. (Specifically, the ith and jth standard basis vectors.) Speaking loosely, this destroys information about the vectors being multiplied by P, so you can't invert P.