It the following matrix isomorphic?

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In summary, the given linear map L is not isomorphic because it is neither an injection nor a surjection. A counterexample for injectivity is (1,1) and (0,1) mapping to the same point, and a counterexample for surjectivity is (0,1) not being hit by any point in the domain. The kernel and image of L are {(x1, x2)^T: x1 = 0 and x2 = 0}, which is the same as the nullspace of L.
  • #1
soopo
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Is the following matrix isomorphic?

Homework Statement


a) Is the following matrix isomorphic?
[tex] L: \Re^{2} \rightarrow \Re^{2} [/tex]

[tex] L(x) = (-x_{2}, 0), where x = (x_{1}, x_{2}) \in \Re^{2}.[/tex]

b)
Define Ker(L) and Im(L).

The Attempt at a Solution


a) I need to show that the L is bijection. I know that it is injection,
because it can be presented as Ax = b. The mapping is also linear, so L is
surjection. Thus, L is bijection and isomorphic.

b)
[tex]Ker(L) = \Re^{2}[/tex]
[tex]Im(L) = \Re^{2}[/tex]

I am not sure is this enough.
 
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  • #2
The mapping is also linear, so L is surjection.

This is incorrect. There are many examples of linear, non-surjective functions. For instance, the projective function. What is the definition of a surjective function? Moreover, what is the definition of an injective function?
 
  • #3
phreak said:
This is incorrect. There are many examples of linear, non-surjective functions. For instance, the projective function. What is the definition of a surjective function? Moreover, what is the definition of an injective function?

Injection: from one to one i.e. each element in the domain is determined once by the element at the codomain
Surjection: all elements in the codomain are determined by those at the domain

This seems to mean that L is not isomorphic. The reason is perhaps that L is not injection as x1 is not determined at the codomain by the domain. That is all elements of the codomain are not determined.
 
  • #4
L is neither an injection nor a surjection. Just cite an example for each. Find two elements in the domain that map to the same image point and find a point in the range that isn't hit by any domain point. Neither is at all hard.
 
  • #5
Dick said:
L is neither an injection nor a surjection. Just cite an example for each. Find two elements in the domain that map to the same image point and find a point in the range that isn't hit by any domain point. Neither is at all hard.

Let x2 = 0 and x1 = 0. This means that two points in the domain map to one codomain. So it is not an injection.

The other one is harder. I first thought to let x2 = 1 and x1 = 1. Then, I started to wonder that it is not even a funtion as no domain points hit points at the codomain.

Please, let me know whether I am at the right direction or not.
 
  • #6
You do mean (x1,x2)->(-x2,0), right? (1,1)->(-1,0), (0,1)->(-1,0), not injective. Nothing maps to (0,1), not surjective. Case closed.
 
  • #7
Let x2 = 0 and x1 = 0. This means that two points in the domain map to one codomain. So it is not an injection.

No, no. To prove that it's not an injection, you must find two points in the domain that map to the same point. The domain is [tex]\mathbb{R}^2[/tex]. Hence, you must find two points x=(x1,x2) and y=(y1,y2) such that [tex]L(x) = L(x_1,x_2) = L(y_1,y_2) = L(y)[/tex]. Since you seem to be stuck, I'll give you an example. Let x= (1,1) and y = (0,1). Then by definition of L, we have L(x) = (-x2,0) = (-1,0) = (1,0) and L(y) = (-y2,0) = (-1,0). This means that we have found two points in the domain of L that map to the same point. But this contradicts the definition of injectivity.
 
  • #8
Dick said:
You do mean (x1,x2)->(-x2,0), right? (1,1)->(-1,0), (0,1)->(-1,0), not injective. Nothing maps to (0,1), not surjective. Case closed.

Thank you!
Your answer is excellent.
 
  • #9
You WILL have an easier time with the next one, right? Don't overthink it. It's not all that abstract.
 
  • #10
b)
The image should be
Im(L) = ([tex]x_{1}, x_{2}[/tex])

The kernel should be got by solving the null space that is Ax = 0.
I am wondering how to put the domain and codomain to Ax = 0.
We should get two simple equations from the matrix.

I am keen on saying that the kernel is
Ker(L) = ([tex]x_{1} = 0, x_{2} = 0[/tex])
It seems to be the only solution - not sure though.
 
  • #11
What linear map are you talking about now?
 
  • #12
Dick said:
What linear map are you talking about now?

I will correct the mistake.

We get the kernel by solving the nullspace
Ker(L) = {[tex](x_{1}, x_{2})^{T}: x_{1} = 0[/tex] and [tex] x_{2} = 0[/tex]}

The image is the same as we apparently want it for the nullspace
Im(L) = {[tex](x_{1}, x_{2})^{T}: x_{1} = 0[/tex] and [tex] x_{2} = 0[/tex]}

It should now be correct.
 
  • #13
Please answer Dick's question! What linear map are you talking about? Are you still talking about L(x1,x2)= (-x2, 0)? If so, then, no, the kernel is NOT {(0,0)}.

And it makes no sense to say of the image "apparently we want it for the nullspace". The image of L is set of all y such that L(x)= y for some x. It has little to do with the nullspace (= kernel).
 

Related to It the following matrix isomorphic?

1. What is an isomorphic matrix?

An isomorphic matrix is a matrix that has the same structure as another matrix, but with different elements. This means that the two matrices have the same number of rows and columns, and the elements are arranged in the same way, but they may have different numerical values.

2. How can you tell if two matrices are isomorphic?

To determine if two matrices are isomorphic, you can compare their dimensions, or the number of rows and columns they have. If the matrices have the same dimensions, then they may be isomorphic, but further analysis is needed to confirm this.

3. Why is it important to know if two matrices are isomorphic?

Knowing if two matrices are isomorphic can help in understanding the relationship between the two matrices. It can also be useful in solving problems related to linear algebra and matrix operations.

4. Can two matrices be isomorphic if they have different dimensions?

No, two matrices cannot be isomorphic if they have different dimensions. In order for two matrices to be isomorphic, they must have the same number of rows and columns.

5. Are isomorphic matrices always equal?

No, isomorphic matrices are not always equal. As mentioned before, isomorphic matrices have the same structure, but different elements. This means that the numerical values in the matrices can be different, making them not equal.

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