Finding inverse of linear mapping

In summary, the conversation discusses finding an inverse for a function f:X->Y where X and Y are normed vector spaces. The person asking the question is unsure of how to find the inverse for their function, as they have only been trained in the Reals and the operations they are familiar with do not apply to vector spaces. The expert explains that if no additional conditions are imposed on f, X, or Y, finding an inverse is impossible. However, if f is bounded and continuous, it is possible to show a homeomorphism and construct a continuous inverse. The expert also mentions that unless there are severe restrictions on X and Y, this is likely the most explicit construction possible.
  • #1
Somefantastik
230
0
so for a mapping f:X->Y

where X,Y are Normed Vector Spaces

if I have a function f(x) = y such that x in X and y in Y, how do I explicitly find f inverse?

I sat down to do this and realize I've only been trained in the Reals where you switch the x,y and then solve for y. But this won't work for my function as my x,y are vectors (not scalars) and I don't have all the operations available like 1/x and ln(x).
 
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  • #2
By a mapping do you mean a bounded linear map? And by inverse do you require your inverse to be bounded linear as well? Maybe you are only working with isometries?

If you do not impose more conditions on f, X or Y than that f is bounded linear, then the problem is impossible in general. Just let X be the trivial normed vector space {0} and let Y be the normed vector space [itex]\mathbb{C}[/itex]. Then we may define [itex]f : X \to Y[/itex] by f(0) = 0. Clearly however f cannot have an inverse because X and Y are not of equal cardinality. The only function [itex]g : Y \to X[/itex] is g(y) = 0, but then f(g(1)) = f(0) = 0 so g is not a right inverse.
 
  • #3
I hope my function is bounded and continuous. My ultimate goal is to show a homeomorphism from a normed vector space to another. So the function I pick must be bicontinuous (and therefore bounded). So let's assume I was smart enough to pick a bicontinuous and bounded function.
 
  • #4
That is sufficient. Since f is a homeomorphism it has a continuous inverse g : Y -> X. You can show fairly easily that g is linear. g is linear and continuous and therefore bounded.

Whether this is explicit enough or not I do not know, but I doubt you will find a much more explicit construction unless you are willing to severely restrict what X and Y are allowed to be.
 
  • #5


Finding the inverse of a linear mapping in normed vector spaces can be a bit more complicated than in the real numbers. However, there are still some general steps that can be followed to find the inverse function.

Firstly, it is important to note that a linear mapping between two normed vector spaces can be represented by a matrix. So, one approach to finding the inverse function would be to find the inverse of this matrix.

Another approach would be to use the concept of the adjoint operator. The adjoint operator of a linear mapping is defined as the transpose of the matrix representing the mapping. So, by finding the adjoint operator, we can effectively find the inverse function.

Additionally, there are some specific properties that can be used to simplify the process of finding the inverse function. For example, if the linear mapping is invertible, then its inverse will also be a linear mapping. This means that we can use techniques such as Gaussian elimination to find the inverse function.

In summary, finding the inverse of a linear mapping in normed vector spaces may require some additional techniques and properties, but it is still possible to determine the inverse function. It may be helpful to consult with a mathematician or refer to additional resources for specific examples and methods.
 

FAQ: Finding inverse of linear mapping

1. What is the definition of a linear mapping?

A linear mapping, also known as a linear transformation, is a function that maps one vector space to another in a way that preserves the operations of addition and scalar multiplication. In other words, for any two vectors u and v and any scalar c, the function satisfies the following properties:
- f(u + v) = f(u) + f(v) (additive property)
- f(cu) = cf(u) (homogeneity property)
Linear mappings are commonly used in mathematics, physics, and engineering to model real-world relationships between quantities.

2. What is the purpose of finding the inverse of a linear mapping?

The inverse of a linear mapping is used to "undo" the effects of the original mapping. This is useful for solving equations, finding the original input values, and understanding the relationship between the input and output of the function. Additionally, the inverse of a linear mapping can be used to transform data from one coordinate system to another.

3. How is the inverse of a linear mapping calculated?

The inverse of a linear mapping can be calculated using matrix operations. Specifically, the inverse of a linear mapping can be found by taking the inverse of the coefficient matrix and multiplying it by the inverse of the constant vector. This results in a new matrix that represents the inverse mapping.

4. Can all linear mappings have an inverse?

No, not all linear mappings have an inverse. In order for a linear mapping to have an inverse, it must be a one-to-one function. This means that each input must correspond to a unique output, and vice versa. If a linear mapping is not one-to-one, then there may be multiple inputs that result in the same output, making it impossible to find the inverse.

5. What are some real-world applications of finding the inverse of a linear mapping?

The inverse of a linear mapping has many practical applications, such as:
- In economics, it can be used to analyze supply and demand relationships.
- In physics, it can be used to model the inverse relationship between force and acceleration.
- In computer graphics, it can be used to transform objects in 3D space.
- In cryptography, it can be used to encrypt and decrypt messages.
- In navigation, it can be used to convert between different coordinate systems.

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