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Math Help - homomorphism

  1. #1
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    homomorphism

    Hi could someonw please help with the following problem:


    homomorphism-untitled.jpg

    I know that for it to be homomorphism;

    A homomorphism from two rings is a function such that

    F(a+b)=F(a)+F(b) and F(a*b)= F(a)*f(b)

    but how do i apply it to this example thanks.
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  2. #2
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    f\left( \begin{bmatrix}{a}&{b}\\{0}&{d}\end{bmatrix}+ \begin{bmatrix}{x}&{y}\\{0}&{z}\end{bmatrix}\right  )=f\left( \begin{bmatrix}{a+x}&{b+y}\\{0}&{d+z}\end{bmatrix}  \right)=a+x=f\left( \begin{bmatrix}{a}&{b}\\{0}&{d}\end{bmatrix} \right)+f\left( \begin{bmatrix}{x}&{y}\\{0}&{z}\end{bmatrix}\right  )

    Try proving the other condition. To compute the kernel: f\left(\begin{bmatrix}{a}&{b}\\{0}&{d}\end{bmatrix  } \right)=0 \Leftrightarrow a=0 \implies ker(f)=\hdots. To compute the image, note that the parameters of each matrix are in \mathbb{R}, then, the parameter a can be any number in \mathbb{R}. Now you can conclude.
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  3. #3
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    Quote Originally Posted by felper View Post
    f\left( \begin{bmatrix}{a}&{b}\\{0}&{d}\end{bmatrix}+ \begin{bmatrix}{x}&{y}\\{0}&{z}\end{bmatrix}\right  )=f\left( \begin{bmatrix}{a+x}&{b+y}\\{0}&{d+z}\end{bmatrix}  \right)=a+x=f\left( \begin{bmatrix}{a}&{b}\\{0}&{d}\end{bmatrix} \right)+f\left( \begin{bmatrix}{x}&{y}\\{0}&{z}\end{bmatrix}\right  )

    Try proving the other condition. To compute the kernel: f\left(\begin{bmatrix}{a}&{b}\\{0}&{d}\end{bmatrix  } \right)=0 \Leftrightarrow a=0 \implies ker(f)=\hdots. To compute the image, note that the parameters of each matrix are in \mathbb{R}, then, the parameter a can be any number in \mathbb{R}. Now you can conclude.

    Thus the image is \mathbb{R}?
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  4. #4
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    Quote Originally Posted by SubZero View Post
    Thus the image is \mathbb{R}?
    Exactly.
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