Inverse functions in R^n

GregA

Sorry if this is the wrong place for my question, I'm having difficulty on a conceptual level getting my head round inverse functions and compositions of functions in R^n. I'm failing to understand my lecture notes as a result.

Suppose I have some function with domain R^n which maps to R^m given by f(x) = f[x¹,x²,...,xⁿ]^T=[f¹(x),f²(x),...,f^m(x)]^T it seems reasonable that you'd want to define f^-1(x) such that f o f^-1(x) = I, but is I an identity matrix?. I ask this because f(x) is a vector in R^m, I'd expect some other function g(f(x)) would also be a vector (as opposed to a matrix).
I'm clearly missing something. Can anyone throw me any hints or direct me to some online material that would help me (I have a book on the way in the post)

Russell Berty

f(f^-1(x)) = x, not simply I.

You might be thinking of Ix which is x.

slider142

Your notation for I is incomplete. Your I is an identity on the image of f (a subset of R^m), while the alternative [itex]J = f^{-1}\circ f[/itex] is an identity on Rⁿ.
Matrices represent linear transformations, not the result of a linear transformation. Ie., if g is a linear transformation from R^m into Rⁿ, then g(x) is a vector in Rⁿ while g can be represented by an nxm matrix.

GregA

aha!...cheers guys! You're right, I was considering f acting on a vector instead of considering that f by itself is an mxn matrix

Things make sense again!