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$T : R^{n\times m}\to R^{m\times m}$, $C \to (AC)^T$

I need to find the inverse of this linear transformation, when A is an invertible matrix... Can anyone help me with this problem?

I am asked to show that the transformation is bijective if A is inversible so I thought about using that a lin. transformation is bijective iff it has an inverse... maybe there is a better way

I realize that A has either a left or right inverse, and that both are a (n x m) matrix.

If B is the left inverse of A, I have figured that $S(X): B(X)^T$ is the inverse of T

Is that enough or do I also have to find a transformation assuming B is the right inverse of A?

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    $\begingroup$ Hint: If $A$ is invertible, it must be square, and $AA' = A'A = I$. Also the transformation $t$ that assigns a matrix its transpose, is invertible (just take the transpose again). $\endgroup$
    – Fryie
    Oct 21, 2015 at 20:24
  • $\begingroup$ But doesn´t A have to be a (m x n) so that the (AC) multiplication is defined? $\endgroup$ Oct 21, 2015 at 20:43
  • $\begingroup$ @Fryie ....and also for the (AC)^T to be an (m x m) matrix? $\endgroup$ Oct 21, 2015 at 20:50
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    $\begingroup$ That's right. I think there must be a mistake in the question. More generally, a linear transformation from $R^{m\cdot n}$ to $R^{m\cdot m}$ cannot be invertible, since the two vector spaces don't have the same dimension. Unless $m = n$, of course. $\endgroup$
    – Fryie
    Oct 21, 2015 at 21:00
  • $\begingroup$ Yes, thank you, I will try using that m=n for T to have an inverse. $\endgroup$ Oct 21, 2015 at 22:01

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