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Thread: Linear maping, maybe?

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

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

    $\displaystyle S(x_1, x_2, x_3) = (x_2 + x_3, x_1)$, $\displaystyle T(x_1, x_2) = (x_2, x_1, x_1 + x_2)$

    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 $\displaystyle S: R^3 -> R^2$ and $\displaystyle T:R^2 -> R^3$ be defined by

    $\displaystyle S(x_1, x_2, x_3) = (x_2 + x_3, x_1)$, $\displaystyle T(x_1, x_2) = (x_2, x_1, x_1 + x_2)$

    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 $\displaystyle SoT(x_1,x_2)$ :

    $\displaystyle T(x_1,x_2)=(x_2, x_1, x_1 + x_2)$

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

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

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

    Hence :

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


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


    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 $\displaystyle m = x_2$ (1) for example...

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

    Dunno if that sounds right the way ive explained it but it works when i try it cos i got $\displaystyle TS = (x_1, x_2 + x_3, x_1 + x_2 + x_3)$ 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 $\displaystyle x_1$ in the equation $\displaystyle (x_2 + x_3, x_1)$ must be replaced by $\displaystyle m = x_2$...
    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 $\displaystyle S(m,n,p)=({\color{red}n+p},{\color{blue}m})$.
    I renamed this on purpose, because it would confuse you.
    Here :
    $\displaystyle m=x_2$ (1)
    $\displaystyle n=x_1$ (2)
    $\displaystyle p=x_1+x_2$ (3)

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