# Prove isomorphism

#### 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.

Swlabr

#### tonio

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

#### Chris L T521

MHF Hall of Fame
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 $$\displaystyle P_3$$?)

#### 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

#### Chris L T521

MHF Hall of Fame
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 $$\displaystyle L:M_{22}\mapsto P_3$$?

#### wattkow

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 $$\displaystyle ax^3+bx^2+cx+d$$ (Speechless)

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

#### Chris L T521

MHF Hall of Fame
I'm not sure how to relate
$$\displaystyle A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$$ with $$\displaystyle ax^3+bx^2+cx+d$$ (Speechless)
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?

#### wattkow

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?

#### Swlabr

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.)

#### 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