Isomorphism math question

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In summary, the conversation discusses isomorphisms and how to prove that a linear map is an isomorphism. The definition of isomorphism is given, and it is mentioned that it should be defined as a map that is both one-to-one and onto. The proof for a linear map being an isomorphism is also given using the properties of linearity.
  • #1
Am I doing this right? I'd appreciate any feedback.

Let T:U ---> v be an isomorphism. Show that T^-1: V----> U is linear.

i. T^-1(0) = 0

ii. T^-1(-V) = -T^-1(V)
T^-1(-0) = T^-1(0+0)
= T^-1(0) + T^-1(0)

T^-1(0) = 0
T^-1(-V) = T^-1((-1)V)
=(-1)T^-1(V)
= -T^-1(V)

If T[x,y,z] = [x-y, y-z, x+z]
Then T is one-to-one right?

How do I show that T is onto?
 
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  • #2
what is your definition of isomorphism?

my definition of isomorphism for a linear map T:U-->V is that there exists a linear map S:V-->U such that SoT and ToS are both the identity maps.

then my version of your statement would be to show that a bijective linear map is an isomorphism, i.e. it has an inverse which is linear.

i.e. to answer you question, your definition of isomorphism probably assumes T is onto.
 
  • #3
mathwonk said:
what is your definition of isomorphism?

my definition of isomorphism for a linear map T:U-->V is that there exists a linear map S:V-->U such that SoT and ToS are both the identity maps.

then my version of your statement would be to show that a bijective linear map is an isomorphism, i.e. it has an inverse which is linear.

i.e. to answer you question, your definition of isomorphism probably assumes T is onto.

An isomorphism is something that is both one-ton-one and onto. But, we were also given that if T: W-->V and S:V-->U are linear transformations, then SoT:W-->U is linear.
 
  • #4
[itex] \hat{T} [/itex] linear:

[tex] \hat{T}(ax+by)=a\hat{T}(x)+b\hat{T}(y) [/tex] (1)

There exists [itex] \hat{T}^{-1} [/itex] so that [itex] \hat{T}^{-1}\left(\hat{T}(x)\right)=\hat{1}x=x [/tex] (2)

Therefore

[tex] \hat{T}^{-1}\left(\hat{T}\left(ax+by\right)\right)=\hat{1}(ax+by)=ax+by [/tex] (3) (by virtue of the definition of isomorphism of vector spaces)

From (1),it follows that

[tex] \hat{T}^{-1}\left(\hat{T}\left(ax+by\right)\right)=T^{-1}\left(a\hat{T}(x)+b\hat{T}(y)\right) [/tex] (4)

Now,again from (2),i write

[tex] ax+by=a\hat{T}^{-1}\left(\hat{T}(x)\right)+b\hat{T}^{-1}\left(\hat{T}(y)\right) [/tex] (5)

Comparing (3),(4) & (5),one gets

[tex] T^{-1}\left(a\hat{T}(x)+b\hat{T}(y)\right)=a\hat{T}^{-1}\left(\hat{T}(x)\right)+b\hat{T}^{-1}\left(\hat{T}(y)\right) [/tex] (6)

Q.e.d.

Daniel.
 
  • #5
by the way, you may not appreciate this for a while, but the word isomorphism should not be defined as your source does, but as i did above. the point is that my definition works for all types of maps, linear, continuous, differentiable, group homomorphism, whatever.

i.e. in many settings a morphism which is one one and onto is still not an isomorphism; e.g. the map taking x to x^3 on the real line does not have a differentiable inverse, but is one one and onto, hence is not a differentiable isomorphism.
 

1. What is isomorphism in mathematics?

Isomorphism is a mathematical concept that describes a relationship between two mathematical structures that preserves the structure and properties of the original structure. In other words, two structures are isomorphic if they have the same algebraic properties and can be transformed into one another.

2. How do you determine if two structures are isomorphic?

To determine if two structures are isomorphic, you must show that there is a one-to-one correspondence between the elements of the two structures that preserves their algebraic properties. This means that for every element in one structure, there is a corresponding element in the other structure that has the same relationships with other elements.

3. Can isomorphism be applied to any mathematical structure?

Yes, isomorphism can be applied to any mathematical structure as long as the structure has defined operations and relationships between its elements. This includes structures such as groups, rings, fields, and vector spaces.

4. What is the significance of isomorphism in mathematics?

Isomorphism is a fundamental concept in mathematics as it allows us to understand the underlying structure and properties of different mathematical objects. It also helps us to simplify and generalize complex mathematical ideas by showing that seemingly different structures are actually equivalent.

5. Are there different types of isomorphism?

Yes, there are different types of isomorphism depending on the type of mathematical structures being compared. For example, group isomorphism, ring isomorphism, and vector space isomorphism are all different types of isomorphism that describe the relationship between different structures.

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