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Math Help - Linear maping, maybe?

  1. #1
    Super Member Deadstar's Avatar
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    Linear maping, maybe?

    Let S: R^3 -> R^2 and T:R^2 -> R^3 be defined by

    S(x_1, x_2, x_3) = (x_2 + x_3, x_1), T(x_1, x_2) = (x_2, x_1, x_1 + x_2)

    Find expressions for ST and TS.

    Any help please?
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  2. #2
    Moo
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    Hello,

    Quote Originally Posted by Deadstar View Post
    Let S: R^3 -> R^2 and T:R^2 -> R^3 be defined by

    S(x_1, x_2, x_3) = (x_2 + x_3, x_1), T(x_1, x_2) = (x_2, x_1, x_1 + x_2)

    Find expressions for ST and TS.

    Any help please?
    Are you looking for the composition of S with T and T with S ? o.O

    Let's see for SoT(x_1,x_2) :

    T(x_1,x_2)=(x_2, x_1, x_1 + x_2)

    ---> SoT(x_1,x_2)=S(T(x_1,x_2))=S(x_2, x_1, x_1 + x_2)

    We know that S(m,n,p)=({\color{red}n+p},{\color{blue}m}).
    I renamed this on purpose, because it would confuse you.
    Here :
    m=x_2 (1)
    n=x_1 (2)
    p=x_1+x_2 (3)

    Therefore {\color{red}n+p}=(2)+(3)=\boxed{2x_1+x_2}
    And \boxed{{\color{blue}m}=x_2}

    Hence :

    S(x_2, x_1, x_1 + x_2)=(2x_1+x_2, \ x_2)


    ---> \boxed{SoT(x_1, \ x_2)=(2x_1+x_2, \ x_2)}


    Is it what you wanted ? Does it help ?
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  3. #3
    Super Member Deadstar's Avatar
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    Yeah i think i get it now.

    I dont know how to explain it exactly...

    Is this right...
    Since m = x_2 (1) for example...

    Any x_1 in the equation (x_2 + x_3, x_1) must be replaced by m = x_2...

    Dunno if that sounds right the way ive explained it but it works when i try it cos i got TS = (x_1, x_2 + x_3, x_1 + x_2 + x_3) which was the right answer!
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  4. #4
    Moo
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    Hmm how to explain... Actually, I replace by m, n and p because it was redundant and really confusing.
    It's equivalent to say "the first coordinate of the image of a triplet by T is the sum of its two last coordinates".

    Quote Originally Posted by Deadstar View Post

    Any x_1 in the equation (x_2 + x_3, x_1) must be replaced by m = x_2...
    It sounds strange to me... Because once you replace with m, n or p, you don't have to say "I'll replace it by...".

    We know that S(m,n,p)=({\color{red}n+p},{\color{blue}m}).
    I renamed this on purpose, because it would confuse you.
    Here :
    m=x_2 (1)
    n=x_1 (2)
    p=x_1+x_2 (3)

    Therefore {\color{red}n+p}=(2)+(3)=\boxed{2x_1+x_2}
    And \boxed{{\color{blue}m}=x_2}
    This part was more for explaining the thing to you than for writing the correct answer.
    You have your own way to explain it. I don't think your teacher would bother that much if you showed it your way. But make it understandable
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