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Math Help - I don't get this question (T-invariant)

  1. #1
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    I don't get this question (T-invariant)

    Create a non-trivial linear operator T with a kernel containing more than the zero vector for which {k (some symmetric matrix) | k belong R} is a t-invariant space in B2 ( symmetric matrices of size 2x2) )


    Would someone please rewrite this question for me in more details, or a better clarification ?
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  2. #2
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    Quote Originally Posted by Revolutionofidentity View Post
    Create a non-trivial linear operator T with a kernel containing more than the zero vector for which {k (some symmetric matrix) | k belong R} is a t-invariant space in B2 ( symmetric matrices of size 2x2) )


    Would someone please rewrite this question for me in more details, or a better clarification ?

    It seems to be like this: denote B_2:=\{A\in M_2(\mathbb{R})\;\;\;M^t=M\;\} , and you have to define a linear operator T:B_2\rightarrow B_2 s.t. \ker T\neq\{0\} ...piece of cake!

    So choose any basis of B_2 and any 2 matrices in M_2(\mathbb{R}) and a third one being the zero matrix, and send the vectors of the basis to these three chosen matrices, and extend by linearity...

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    It seems to be like this: denote B_2:=\{A\in M_2(\mathbb{R})\;\;\;M^t=M\;\} , and you have to define a linear operator T:B_2\rightarrow B_2 s.t. \ker T\neq\{0\} ...piece of cake!

    So choose any basis of B_2 and any 2 matrices in M_2(\mathbb{R}) and a third one being the zero matrix, and send the vectors of the basis to these three chosen matrices, and extend by linearity...

    Tonio

    Hi thanks a lot for the reply. it helped but i'm confused about something, in your guidelines i don't see where i'm gonna use my t-invariant space, it was given in specific numbers.
    K[[3,7],[7,-4]], that was a t-invariant in B2.
    And by the way, how to "Thank" you ? i don't know how to do that .
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  4. #4
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    Quote Originally Posted by Revolutionofidentity View Post
    And by the way, how to "Thank" you ? i don't know how to do that .
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  5. #5
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    Quote Originally Posted by Revolutionofidentity View Post
    Hi thanks a lot for the reply. it helped but i'm confused about something, in your guidelines i don't see where i'm gonna use my t-invariant space, it was given in specific numbers.
    K[[3,7],[7,-4]], that was a t-invariant in B2.
    And by the way, how to "Thank" you ? i don't know how to do that .

    Ok, so then the map is from M_2(\mathbb{R}) to itself, and then do as before: choose a basis \{A_1,A_2,A_3,A_4\} and any 3 matrices S_1,S_2,S_3\in B_2 , and map as follows:

    A_i\mapsto S_i\,,\,\,i=1,2,3\,,\,\,A_4\mapsto 0= the zero matrix , and extend this by linearity.

    This maps M_2(\mathbb{R}) to B_2; if you want a more general thing do choose a basis for B_2 , extend it to a basis of all M_2(\mathbb{R}) and map as above when you map the basis of B_2 to 2 matrices of B_2 and to the zero matrix .

    Tonio
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  6. #6
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    Quote Originally Posted by tonio View Post
    It seems to be like this: denote B_2:=\{A\in M_2(\mathbb{R})\;\;\;M^t=M\;\} , and you have to define a linear operator T:B_2\rightarrow B_2 s.t. \ker T\neq\{0\} ...piece of cake!

    So choose any basis of B_2 and any 2 matrices in M_2(\mathbb{R}) and a third one being the zero matrix, and send the vectors of the basis to these three chosen matrices, and extend by linearity...

    Tonio

    Hi Man, it worked !! Thanks a lot, and i picked my t-invariant space to be the random M2 matrix. picked a basis, and picked a B2 matrix that's not trivial, and made it belong to the kernel of T. And Made a linear combination then used axioms of L transformation and it worked. I checked by throwing a random symmetric 2x2 matrix, then by throwing a matrix from the T-invariant space, and the transformation belonged to it.

    Thanks !

    but i have a question, why does it work ???
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  7. #7
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    thanks to God i could solve that question, i really need to pass this assignment in order to pass the whole class.
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  8. #8
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    Quote Originally Posted by Revolutionofidentity View Post
    thanks to God i could solve that question, i really need to pass this assignment in order to pass the whole class.

    "Thanx to god"? Hmmm....anyway, it works because we applied a very basic and important theorem in finite dimensional linear algebra:

    Theorem: If \{v_1,\ldots,v_n\} is a basis of a linear space V , and if \{w_1,\ldots,w_n\} are ANY n vectors in some linear space W defined over the same field as V , then

    there exists a unique linear transformation T:V\rightarrow W s.t. Tv_1=w_i\,\,\,\forall\,i=1,\ldots,n

    Tonio
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