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Thread: Proving that a transformation is surjective (or not)

  1. #1
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    Proving that a transformation is surjective (or not)

    I've got two questions on a handout that we didn't cover in class...

    I guess I'm not understanding what it means to be surjective..

    on 5a, would the first blanks be Let Av be in the codomain R^2?

    https://gyazo.com/daeb403401c00c164ecb7b876a03a1fc
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  2. #2
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    Re: Proving that a transformation is surjective (or not)

    Quote Originally Posted by MrJank View Post
    I've got two questions on a handout that we didn't cover in class...
    I guess I'm not understanding what it means to be surjective..
    Surjective is a fancy word for onto.
    So for any point $\displaystyle \left[ {\begin{array}{*{20}{c}} a\\ b \end{array}} \right]$ you must show that there is a point $\displaystyle \left[ {\begin{array}{*{20}{c}} x\\ y \end{array}} \right]$ such that $\displaystyle T_A\left[ {\begin{array}{*{20}{c}} x\\ y \end{array}} \right]$=$\displaystyle \left[ {\begin{array}{*{20}{c}} a\\ b \end{array}} \right]$
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    Re: Proving that a transformation is surjective (or not)

    Quote Originally Posted by Plato View Post
    Surjective is a fancy word for onto.
    So for any point $\displaystyle \left[ {\begin{array}{*{20}{c}} a\\ b \end{array}} \right]$ you must show that there is a point $\displaystyle \left[ {\begin{array}{*{20}{c}} x\\ y \end{array}} \right]$ such that $\displaystyle T_A\left[ {\begin{array}{*{20}{c}} x\\ y \end{array}} \right]$=$\displaystyle \left[ {\begin{array}{*{20}{c}} a\\ b \end{array}} \right]$
    Okay, so in the context of my problem what would the beginning of a proof look like?
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    Re: Proving that a transformation is surjective (or not)

    Quote Originally Posted by MrJank View Post
    Okay, so in the context of my problem what would the beginning of a proof look like?
    Let $\vec{v}$ be in the codomain of $T_A$.

    Or

    Let $\vec v$ be in the codomain $\mathbb{R}^2$
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    Re: Proving that a transformation is surjective (or not)

    Quote Originally Posted by MrJank View Post
    Okay, so in the context of my problem what would the beginning of a proof look like?
    $\displaystyle T_A\left[ {\begin{array}{*{20}{c}} x\\ y \end{array}} \right]=\left[ {\begin{array}{*{20}{rr}} 2x+y\\ -4x+2y \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} a\\ b \end{array}} \right]$
    Now YOU solve for $x~\&~y$. Show us your work!
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