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## Homework Statement

Y = AX = g(X)

Where X,Y are elements of R^n and A is a nxn matrix.

What is the Jacobian of this transformation, Jg(x)?

## Homework Equations

N.A.

## The Attempt at a Solution

Well, I know what to do in the non-matrix case. For example...

U = g(x,y)

V = h(x,y)

The transformation can be seen as a vector valued function f(x,y) = (g(x,y),h(x,y)). So the jacobian of this transform, Jf(x,y) = the determinant of a matrix with row 1 = [dg/dx , dh/dx] and row 2 = [dg/dy, dh/dy].

So Jf(x,y) = (dg/dx)(dh/dy) - (dg/dy)(dh/dx).

But what do I do in the matrix case? I know g(X) can be seen as a vector with functions as entries but does this help?

Y = AX = g(X) = (g1(X),g2(X),...,gn(X))

Thanks!