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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.

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    $\begingroup$ Hint: Have you used the assumption that $A$ is inventively? $\endgroup$ Commented Apr 24, 2017 at 20:36
  • $\begingroup$ 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? $\endgroup$
    – Thomas
    Commented Apr 24, 2017 at 20:40
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    $\begingroup$ Yes, that's all you need. $\endgroup$ Commented Apr 24, 2017 at 20:41
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    $\begingroup$ @HansHüttel "inventively" :-) $\endgroup$ Commented Apr 24, 2017 at 20:53
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    $\begingroup$ Autocorrect is the devil! $\endgroup$ Commented Apr 24, 2017 at 20:55

1 Answer 1

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It should be a bijective homomorphism.

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 $$ L_{A^{-1}} \circ L_A = \text{id} $$

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