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Math Help - Prove isomorphism

  1. #1
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    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.

    Advathanksnce!
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  2. #2
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    Quote Originally Posted by wattkow View Post
    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
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  3. #3
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by wattkow View Post
    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 A\in M_{22} with A=\begin{bmatrix}a&b\\c&d\end{bmatrix}. What is the most basic mapping L we can define such that L:M_{22}\mapsto P_3? (What do elements look like in P_3?)
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  4. #4
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    What do elements look like in P_3 ?
    ax^3+bx^2+cx+d?

    As you can tell from my answer, I am rather lost in class
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  5. #5
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by wattkow View Post
    What do elements look like in P_3 ?
    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 L such that L:M_{22}\mapsto P_3?
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  6. #6
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    Quote Originally Posted by Chris L T521 View Post
    Be a little more confident! (That is correct, by the way )

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

    Is it just <br />
T(\begin{bmatrix}a&b\\c&d\end{bmatrix})<br />
= <br />
ax^3+bx^2+cx+d<br />
?
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  7. #7
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by wattkow View Post
    I'm not sure how to relate
    <br />
A=\begin{bmatrix}a&b\\c&d\end{bmatrix}<br />
with <br />
ax^3+bx^2+cx+d<br />
    That's how you define the mapping! L:M_{22}\mapsto P_3 where 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?
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  8. #8
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    Quote Originally Posted by Chris L T521 View Post
    That's how you define the mapping! L:M_{22}\mapsto P_3 where 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?
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  9. #9
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by wattkow View Post
    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.)
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  10. #10
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    <br /> <br />
L\left(\begin{bmatrix}a&b\\c& d\end{bmatrix}\right) <br /> <br />
\left(\begin{bmatrix}x\\y\\z\end{bmatrix}\right)<br />
= 0?

    I'm lost because I know it's not possible to multiply a 2x2 matrix by a 3x2 matrix
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  11. #11
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by wattkow View Post
    <br /> <br />
L\left(\begin{bmatrix}a&b\\c& d\end{bmatrix}\right) <br /> <br />
\left(\begin{bmatrix}x\\y\\z\end{bmatrix}\right)<br />
= 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,

    L(M)=0 \Rightarrow \ldots.

    This is basically asking you if ax^3+bx^2+cx+d = 0 then what are a, b, c and d?
    Last edited by Swlabr; May 17th 2010 at 09:19 AM. Reason: Change of notation to keep it consistant. Who uses L for a linear map? It's always T!
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  12. #12
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Swlabr View Post
    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,

    T(M)=0 \Rightarrow \ldots.

    This is basically asking you if ax^3+bx^2+cx+d = 0 then what are a, b, c and d?
    You mean 2x2 matrix...
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  13. #13
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    Quote Originally Posted by Swlabr View Post
    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,

    T(M)=0 \Rightarrow \ldots.

    This is basically asking you if ax^3+bx^2+cx+d = 0 then what are a, b, c and d?
    Is it suppose to look like this:
    <br /> <br />
L\left(\begin{bmatrix}a&b&0\\c&d&0\end{bmatrix}\ri  ght)<br />
?
    I assume a, b, c, d = 0?
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  14. #14
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Chris L T521 View Post
    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.
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  15. #15
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by wattkow View Post
    Is it suppose to look like this:
    <br /> <br />
L\left(\begin{bmatrix}a&b&0\\c&d&0\end{bmatrix}\ri  ght)<br />
?
    I assume 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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