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Math Help - Matrix proofs

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

    Let T : \Re \rightarrow \Re ^3 be given by T \begin{pmatrix}<br />
{x}\\ <br />
{y}\\ <br />
{z}<br />
\end{pmatrix}=\begin{pmatrix}<br />
{x}\\ <br />
{y}\\ <br />
{0}<br />
\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:

    \begin{pmatrix}<br />
{1}&{0}&{0}\\ <br />
{0}&{1}&{0}\\ <br />
{0}&{0}&{0}<br />
\end{pmatrix}\begin{pmatrix}<br />
{x}\\ <br />
{y}\\ <br />
{z}<br />
\end{pmatrix}=\begin{pmatrix}<br />
{x}\\ <br />
{y}\\ <br />
{0}<br />
\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:

    \begin{pmatrix}<br />
{1}&{0}&{0}\\ <br />
{0}&{1}&{0}\\ <br />
{0}&{0}&{0}<br />
\end{pmatrix}\begin{pmatrix}<br />
{x}\\ <br />
{y}\\ <br />
{z}<br />
\end{pmatrix}=\begin{pmatrix}<br />
{x}\\ <br />
{y}\\ <br />
{0}<br />
\end{pmatrix}

    ?

    Help would be appreciated!
    Yes!
    Since T: \mathbb{R}^3 \to \mathbb{R}^3 is a linear transformation it means T(\bold{x}) = A\bold{x}.
    Where A is a 3\times 3 matrix.
    That is the associated matrix with 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 T is a linear transformation you need to show that: T(\bold{x}+\bold{y}) = T(\bold{x}) + T(\bold{y}) and 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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