Are inverse functions in R^n an identity matrix?

In summary, the conversation discusses the difficulty with understanding inverse functions and compositions of functions in R^n. It is suggested that defining f^-1(x) such that f o f^-1(x) = I could be useful, but the identity matrix I may not be the correct choice. The conversation concludes with the realization that f by itself is an mxn matrix and not just a function acting on a vector.
  • #1
GregA
210
0
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)
 
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  • #2
f(f^-1(x)) = x, not simply I.

You might be thinking of Ix which is x.
 
  • #3
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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  • #4
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!
 

1. What are inverse functions in R^n?

Inverse functions in R^n are functions that "undo" the original function. In other words, if a function maps a set of inputs to a set of outputs, the inverse function maps those outputs back to the original inputs. This is commonly represented as f^{-1}(x), where x is the input variable.

2. How do you find the inverse function of a given function in R^n?

To find the inverse function of a given function in R^n, you can follow a few steps. First, rewrite the function in the form y = f(x). Then, switch the variables x and y. Finally, solve for y to get the inverse function. It is important to note that not all functions have an inverse function.

3. What is the domain and range of an inverse function in R^n?

The domain of an inverse function in R^n is the range of the original function, and the range of an inverse function is the domain of the original function. This is because the input and output values are switched in the inverse function. However, it is important to note that the domain and range of an inverse function may be restricted due to the original function's domain and range.

4. How do you test if two functions in R^n are inverses of each other?

To test if two functions in R^n are inverses of each other, you can use the composition of functions. If the composition of the two functions results in the input value, then they are inverses. This can be represented as (f \circ g)(x) = x, where f and g are the two functions being tested.

5. What is the importance of inverse functions in R^n?

Inverse functions in R^n are important in many areas of mathematics and science. For example, they are used in calculus to find derivatives and integrals, and in statistics to model relationships between variables. They also have applications in engineering, physics, and computer science. Understanding inverse functions is crucial in problem-solving and analyzing real-life situations.

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