Prove isomorphism

May 2010
20
1
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.

Advathanksnce!
 
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Oct 2009
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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.

Advathanksnce!

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
May 2008
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Chicago, IL
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.

Advathanksnce!
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 \(\displaystyle P_3\)?)
 
May 2010
20
1
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 :eek:
 

Chris L T521

MHF Hall of Fame
May 2008
2,844
2,046
Chicago, IL
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 :eek:
Be a little more confident! (That is correct, by the way :) )

With this, how would you define the mapping \(\displaystyle L\) such that \(\displaystyle L:M_{22}\mapsto P_3\)?
 
May 2010
20
1
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
May 2008
2,844
2,046
Chicago, IL
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?
 
May 2010
20
1
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?
 
May 2009
1,176
412
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.)
 
May 2010
20
1
\(\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 :eek: