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Math Help - Linear Algebra

  1. #1
    Junior Member
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    Exclamation Linear Algebra

    Is anyone able to help me do the following proof please:

    Let f:R3 > R3 be given by

    f(x1,x2,x3)=(x3,x1+x2,x1-x2)

    Prove that f is a linear transformation.


    Many thanks to anyone who can help.
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hi

    You have to prove two things :
    • For all x=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix},\,y=\b  egin{pmatrix}y_1\\y_2\\y_3\end{pmatrix} \in \mathbb{R}^3, f(x+y)=f(x)+f(y)
    • For all \lambda \in \mathbb{R}, \,x=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}\in\m  athbb{R}^3, f(\lambda x)=\lambda f(x)
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  3. #3
    Lord of certain Rings
    Isomorphism's Avatar
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    Quote Originally Posted by flyingsquirrel View Post
    Hi

    You have to prove two things :
    • For all x=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix},\,y=\b  egin{pmatrix}y_1\\y_2\\y_3\end{pmatrix} \in \mathbb{R}^3, f(x+y)=f(x)+f(y)
    • For all \lambda \in \mathbb{R}, \,x=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}\in\m  athbb{R}^3, f(\lambda x)=\lambda f(x)
    Or obtain a matrix A such that f(x) = Ax(You should know the result that this is always a linear map).

    If x=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}\in\mat  hbb{R}^3, \,\, A = \begin{pmatrix}0&0&1\\1&1&0\\1&-1&0\end{pmatrix}
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