1. ## Prove isomorphism

I have dosed off in class way too many times and I need help with a few problems from my pre-test:

#3
Prove that M_22 is isomorphic to P_3.

A simple hint in the right direction would be greatly appreciated.

2. Originally Posted by wattkow
I have dosed off in class way too many times and I need help with a few problems from my pre-test:

#3
Prove that M_22 is isomorphic to P_3.

A simple hint in the right direction would be greatly appreciated.

Hint and very important theorem: any two vector spaces of the same dimension defined over the same field are isomorphic.

Proof: very simple and exists in every decent text book in linear algebra. Look for it.

Tonio

3. Originally Posted by wattkow
I have dosed off in class way too many times and I need help with a few problems from my pre-test:

#3
Prove that M_22 is isomorphic to P_3.

A simple hint in the right direction would be greatly appreciated.

Let $\displaystyle A\in M_{22}$ with $\displaystyle A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$. What is the most basic mapping $\displaystyle L$ we can define such that $\displaystyle L:M_{22}\mapsto P_3$? (What do elements look like in $\displaystyle P_3$?)

4. What do elements look like in $\displaystyle P_3$ ?
$\displaystyle ax^3+bx^2+cx+d$?

As you can tell from my answer, I am rather lost in class

5. Originally Posted by wattkow
What do elements look like in $\displaystyle P_3$ ?
$\displaystyle ax^3+bx^2+cx+d$?

As you can tell from my answer, I am rather lost in class
Be a little more confident! (That is correct, by the way )

With this, how would you define the mapping $\displaystyle L$ such that $\displaystyle L:M_{22}\mapsto P_3$?

6. Originally Posted by Chris L T521
Be a little more confident! (That is correct, by the way )

With this, how would you define the mapping $\displaystyle L$ such that $\displaystyle L:M_{22}\mapsto P_3$?
I'm not sure how to relate
$\displaystyle A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ with $\displaystyle ax^3+bx^2+cx+d$

Is it just $\displaystyle T(\begin{bmatrix}a&b\\c&d\end{bmatrix})$ = $\displaystyle ax^3+bx^2+cx+d$ ?

7. Originally Posted by wattkow
I'm not sure how to relate
$\displaystyle A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ with $\displaystyle ax^3+bx^2+cx+d$
That's how you define the mapping! $\displaystyle L:M_{22}\mapsto P_3$ where $\displaystyle L\left(\begin{bmatrix}a&b\\c& d\end{bmatrix}\right)=ax^3+bx^2+cx+d$.

Can you justify that this mapping is an isomorphism?

8. Originally Posted by Chris L T521
That's how you define the mapping! $\displaystyle L:M_{22}\mapsto P_3$ where $\displaystyle L\left(\begin{bmatrix}a&b\\c& d\end{bmatrix}\right)=ax^3+bx^2+cx+d$.

Can you justify that this mapping is an isomorphism?
Somehow show that it is 1-to-1?

9. Originally Posted by wattkow
Somehow show that it is 1-to-1?
Find the kernel. A linear map is 1-1 if and only if the kernel is trivial.

(That is, find every element which is mapped to zero.)

10. $\displaystyle L\left(\begin{bmatrix}a&b\\c& d\end{bmatrix}\right)$ $\displaystyle \left(\begin{bmatrix}x\\y\\z\end{bmatrix}\right)$ = 0?

I'm lost because I know it's not possible to multiply a 2x2 matrix by a 3x2 matrix

11. Originally Posted by wattkow
$\displaystyle L\left(\begin{bmatrix}a&b\\c& d\end{bmatrix}\right)$ $\displaystyle \left(\begin{bmatrix}x\\y\\z\end{bmatrix}\right)$ = 0?

I'm lost because I know it's not possible to multiply a 2x2 matrix by a 3x2 matrix
This isn't what you are trying to do. You want to find the kernel of the map L, which will be a 4-by-4 matrix but you don't want to think of it that way. You need to solve,

$\displaystyle L(M)=0 \Rightarrow \ldots$.

This is basically asking you if $\displaystyle ax^3+bx^2+cx+d = 0$ then what are $\displaystyle a, b, c$ and $\displaystyle d$?

12. Originally Posted by Swlabr
This isn't what you are trying to do. You want to find the kernel of the map T, which will be a 4-by-4 matrix but you don't want to think of it that way. You need to solve,

$\displaystyle T(M)=0 \Rightarrow \ldots$.

This is basically asking you if $\displaystyle ax^3+bx^2+cx+d = 0$ then what are $\displaystyle a, b, c$ and $\displaystyle d$?
You mean 2x2 matrix...

13. Originally Posted by Swlabr
This isn't what you are trying to do. You want to find the kernel of the map T, which will be a 4-by-4 matrix but you don't want to think of it that way. You need to solve,

$\displaystyle T(M)=0 \Rightarrow \ldots$.

This is basically asking you if $\displaystyle ax^3+bx^2+cx+d = 0$ then what are $\displaystyle a, b, c$ and $\displaystyle d$?
Is it suppose to look like this:
$\displaystyle L\left(\begin{bmatrix}a&b&0\\c&d&0\end{bmatrix}\ri ght)$ ?
I assume $\displaystyle a, b, c, d = 0$?

14. Originally Posted by Chris L T521
You mean 2x2 matrix...
No, 4-by-4. You are mapping from a 4-dimensional vector space into a 4-dimensional vector space...

The matrix you are mapping from is 2-by-2, and I was presuming the OP was getting confused by the linear map ~ matrix equivalence (it seemed like he was trying to find the kernel of the 2-by-2 matrix). That is why I wanted to point out that T is 4-by-4.

15. Originally Posted by wattkow
Is it suppose to look like this:
$\displaystyle L\left(\begin{bmatrix}a&b&0\\c&d&0\end{bmatrix}\ri ght)$ ?
I assume $\displaystyle a, b, c, d = 0$?
Is what supposed to look like that?

(The map T in my earlier post is the same as the L in the rest of the thread. I have changed this to reflect that.)

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