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Thread: Matrix proofs

  1. #1
    Super Member Showcase_22's Avatar
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    Matrix proofs

    Let $\displaystyle T : \Re \rightarrow \Re ^3$ be given by $\displaystyle T \begin{pmatrix}
    {x}\\
    {y}\\
    {z}
    \end{pmatrix}=\begin{pmatrix}
    {x}\\
    {y}\\
    {0}
    \end{pmatrix}$. This is projection from 3-space onto the xy-plane. Show that T is a linear transformation. What is the matrix associated to T?
    I'm not really sure how to prove this is a linear transformation. I couldn't find a definition for a linear transformation either which made the problem more confusing.

    I also don't know what it means by "matrix associated to T". Is this all it wants:

    $\displaystyle \begin{pmatrix}
    {1}&{0}&{0}\\
    {0}&{1}&{0}\\
    {0}&{0}&{0}
    \end{pmatrix}\begin{pmatrix}
    {x}\\
    {y}\\
    {z}
    \end{pmatrix}=\begin{pmatrix}
    {x}\\
    {y}\\
    {0}
    \end{pmatrix}$

    ?

    Help would be appreciated!
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  2. #2
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    Quote Originally Posted by Showcase_22 View Post
    I'm not really sure how to prove this is a linear transformation. I couldn't find a definition for a linear transformation either which made the problem more confusing.

    I also don't know what it means by "matrix associated to T". Is this all it wants:

    $\displaystyle \begin{pmatrix}
    {1}&{0}&{0}\\
    {0}&{1}&{0}\\
    {0}&{0}&{0}
    \end{pmatrix}\begin{pmatrix}
    {x}\\
    {y}\\
    {z}
    \end{pmatrix}=\begin{pmatrix}
    {x}\\
    {y}\\
    {0}
    \end{pmatrix}$

    ?

    Help would be appreciated!
    Yes!
    Since $\displaystyle T: \mathbb{R}^3 \to \mathbb{R}^3$ is a linear transformation it means $\displaystyle T(\bold{x}) = A\bold{x}$.
    Where $\displaystyle A$ is a $\displaystyle 3\times 3$ matrix.
    That is the associated matrix with $\displaystyle T$.
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  3. #3
    Super Member Showcase_22's Avatar
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    Awesome!

    but how do I show it's a linear transformation?

    Can I just explain that (x,y,0) is a line since it has only one dimension (length)?
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  4. #4
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    Quote Originally Posted by Showcase_22 View Post
    but how do I show it's a linear transformation?
    To show that $\displaystyle T$ is a linear transformation you need to show that: $\displaystyle T(\bold{x}+\bold{y}) = T(\bold{x}) + T(\bold{y})$ and $\displaystyle T(k\bold{x}) = kT(\bold{x})$.
    Can you do those two steps?
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  5. #5
    Super Member Showcase_22's Avatar
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    yes I can (and have!)

    Cheers! That's some homework I won't have to do at the weekend! =D
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