Is the linear transformation y = A(Bx) invertible?

**centenial** · March 1st 2011, 02:59 PM

I'm having trouble with this question.

Here is what I have:

From the hint, we solve the equation $\vec{y}=A(B\vec{x})$ first for $B(\vec{x})$ and then for $\vec{x}$ . So we have

$B\vec{x}=A^{-1}\vec{y}$

so...

$\vec{x}=B^{-1}A^{-1}\vec{y}$

I'm really not sure where to go from here though.

**Ackbeet** · March 1st 2011, 06:30 PM

Well, what does the fact that both $A$ and $B$ are invertible (so you can write their inverses) say about the last equation you wrote down? Is the transformation invertible?

**centenial** · March 1st 2011, 07:01 PM

Ok, I think I understand.

So it is invertible and its inverse is simply:

$\vec{x} = B^{-1}A^{-1}\vec{y}$

Is that correct?

**Ackbeet** · March 1st 2011, 07:10 PM

I suppose your answer is in keeping with the question's nomenclature in the OP. I would say that $AB$ is the linear transformation, and that

$(AB)^{-1}=B^{-1}A^{-1}$

is the inverse transformation.

I think you get the main mathematical idea, though.

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