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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- May 17th 2010, 12:42 AMwattkowProve 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.

Advathanksnce! - May 17th 2010, 01:26 AMtonio
- May 17th 2010, 07:05 AMChris L T521
- May 17th 2010, 07:19 AMwattkow
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 :o - May 17th 2010, 07:25 AMChris L T521
- May 17th 2010, 07:33 AMwattkow
- May 17th 2010, 07:41 AMChris L T521
- May 17th 2010, 07:49 AMwattkow
- May 17th 2010, 07:50 AMSwlabr
- May 17th 2010, 08:04 AMwattkow
$\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 :o - May 17th 2010, 08:09 AMSwlabr
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 $\displaystyle d$? - May 17th 2010, 08:12 AMChris L T521
- May 17th 2010, 08:15 AMwattkow
- May 17th 2010, 08:16 AMSwlabr
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. - May 17th 2010, 08:16 AMSwlabr