Are inverse functions in R^n an identity matrix?

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Discussion Overview

The discussion revolves around the conceptual understanding of inverse functions and compositions of functions in the context of R^n and R^m. Participants explore the relationship between a function and its inverse, particularly questioning the nature of the identity matrix in this context.

Discussion Character

  • Conceptual clarification, Technical explanation

Main Points Raised

  • One participant expresses confusion about the definition of the inverse function and its relationship to the identity matrix, questioning whether the identity is indeed a matrix.
  • Another participant clarifies that the composition of a function and its inverse results in the original input, not simply an identity matrix.
  • A further contribution points out that the identity referred to is specific to the image of the function, distinguishing between the identity on the image and the identity on the domain.
  • One participant acknowledges their misunderstanding, realizing that they were misinterpreting the function's representation as a matrix rather than as a transformation.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the initial confusion regarding the identity matrix, but there is a general agreement on the clarification of the function's representation and the nature of compositions.

Contextual Notes

There are limitations in the participants' understanding of the notation and the distinction between functions and their matrix representations, which may affect their interpretations of the concepts discussed.

Who May Find This Useful

This discussion may be useful for students grappling with the concepts of inverse functions, compositions, and the relationship between functions and their matrix representations in linear algebra.

GregA
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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),...,fm(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)
 
Last edited:
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f(f^-1(x)) = x, not simply I.

You might be thinking of Ix which is x.
 
Your notation for I is incomplete. Your I is an identity on the image of f (a subset of Rm), 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 Rm into Rn, then g(x) is a vector in Rn while g can be represented by an nxm matrix.
 
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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 :redface: :smile:

Things make sense again!
 

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