- #1

karnten07

- 213

- 0

## Homework Statement

Let B be an invertible n x n matrix. Prove that the linear transformation L: Mn,n [tex]\rightarrow[/tex] Mn,n given by L(A) = AB, is an isomorphism.

## The Attempt at a Solution

I know to show it is an isomorphism i need to show that L is both onto and one-to-one.

By the theorem that says:

Let T:V[tex]\rightarrow[/tex]W be a linear transformation with vector spaces V and W both of dimension n. Then T is one-to-one if and only if it is onto.

To prove both conditions needed for an isomorphism i can just prove it is one-to-one as in this case, 'V' and 'W' are the same dimension and so proving L is one-to-one also proves it is onto.

*To prove it is one-to-one, i need to determine the kernel of L and show that it is {0}. To do this i need to use the fact that B is an invertible n x n matrix and L(A)=AB.*

I need some guidance on how to use these features to show the kernel of L os {0}?

Thanks in advance