Suppose a linear mapping F: V →U is one-to-one and onto. Show that the inverse mapping F^-1: U →

Suppose a linear mapping F: V →U is one-to-one and onto....

Key Concepts

Linearity of the Inverse Mapping

The concept here is to prove that the inverse of a linear, bijective mapping is also linear. This involves demonstrating that the inverse mapping preserves vector addition and scalar multiplication. The argument generally uses properties inherited from the original mapping, showing that the structure of the vector space is maintained under the inverse mapping, thereby confirming its linear nature.

Inverse Mapping

An inverse mapping reverses the effect of a function such that applying the original function followed by its inverse (or vice versa) results in the original value. In the setting of linear mappings, if the mapping is both one-to-one and onto, an inverse function exists that retrieves the original vector from its image, playing an essential role in solving equations and understanding vector space isomorphisms.

Linear Mapping

A linear mapping is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. This means that for any vectors and any scalars, the function respects these operations, which is fundamental to understanding the structure and behavior of vector spaces in linear algebra.

Injectivity and Surjectivity

In the context of linear mappings, injectivity (one-to-one) ensures that distinct elements in the domain map to distinct elements in the codomain, while surjectivity (onto) guarantees that every element in the codomain is the image of some element from the domain. Both conditions together imply that the mapping is invertible, which is crucial for defining a valid inverse mapping.

Transcript

00:01 Okay, so for this exercise we got a linear mapping f that is bijective, that means it's one to one and on two.

00:08 So we need to show that based on this, the inverse of this function is also linear.

00:18 Okay, so for this we're going to consider two vectors on each vector space.

00:24 So v2 on v1 and u2 on u1, u2.

00:31 And u and even more we're going to say that f of v1 is equals to u1 and f of b2 is equal to u2 and the scalers the scaler in this case alpha on the field okay so first we know that f is linear, is a linear mapping.

01:16 So based on that, f of b1 plus b2 is equal to f of b1 plus f of b2.

01:30 And we know that this is equal to u1 plus u2.

01:39 And also, so this is the first fact because f is linear.

01:47 And the second one is that if we consider f of f, alpha b1 let's say this is equal to alpha f of b1 which at the same time is equal to alpha u1 so this is what we know because f is a linear mapping but now let's define what happened with the inverse so the inverse is a map from u to b and because is the inverse of f, then f inverse of u1, it's going to be equals to v i, v1.

02:41 And the inverse of u2 is going to be v2.

02:53 So what happened if we take f inverse of u1 plus u2? well, we have that because f is linear, then f of u plus v1 plus v2 is equal to u1 plus u2.

03:17 If you apply the inverse to this result, we're going to obtain b1 plus b2...

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