# Show that the linear transformation is an isomorphism

Hi im having trouble with this homework question

"Let $F$ be a field.

Let $A ∈ M_{n×n}(F)$ be an invertible matrix. Show that the linear transformation $L_A : F^ n → F^ n , x→ Ax,$ is an isomorphism."

So since I need to show it's an isomorphism I need to show it's a linear transformation (which is is) along with injectivity and surjectivity. I have shown injectivity quite easily but I'm not sure how to show surjectivity.

Any help would be very much appreciated.

• Hint: Have you used the assumption that $A$ is inventively? Commented Apr 24, 2017 at 20:36
• I was going to set an equation up like $Ax=y$ then pre-multiply both sides by $A^{-1}$ which leaves $x=A^{-1}y$. Would this be along the right lines? Commented Apr 24, 2017 at 20:40
• Yes, that's all you need. Commented Apr 24, 2017 at 20:41
• @HansHüttel "inventively" :-) Commented Apr 24, 2017 at 20:53
• Autocorrect is the devil! Commented Apr 24, 2017 at 20:55

We have $$L_A(x + y) = A (x + y) = Ax + Ay = L_A(x) + L_A(y)$$ so it is a homomorphism.
An invertible matrix satisfies $$A A^{-1} = A^{-1} A = I$$ we have $$x = A A^{-1} x = L_A(A^{-1}x) = L_A(L_{A^{-1}}(x))$$ or $$L_A \circ L_{A^{-1}} = \text{id}$$ and similar $$L_{A^{-1}} \circ L_A = \text{id}$$