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Math Help - Matrices and linear transformations

  1. #1
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    Matrices and linear transformations

    Identify the following matrices as reflections, shears, scalings,rotation or projections for the unit square.

    a) \left(\begin{array}{cc}1/5&2/5\\2/5&4/5\end{array}\right)

    b) \left(\begin{array}{cc}1/2&\sqrt3/2\\-\sqrt3/2&1/2\end{array}\right)

    I thought that b was a straightforward rotation using the rotation matrix :
    \left(\begin{array}{cc}cos\theta&sin\theta\\-sin\theta&cos\theta\end{array}\right)

    where the angle is 60 degrees, but when i graphed the unit square under this transformation it doesnt look like a simple rotation.

    Part a i am totally stuck on.
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  2. #2
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    Quote Originally Posted by ulysses123 View Post
    Identify the following matrices as reflections, shears, scalings,rotation or projections for the unit square.

    a) \left(\begin{array}{cc}1/5&2/5\\2/5&4/5\end{array}\right)
    Notice that <1, 0> is mapped into <1/5, 2/5> and <0, 1> is mapped into <2/5, 4/5>= 2<1/5, 2/5>. That is, both axes are mapped into the single line y= 2x. What does that tell you?

    b) \left(\begin{array}{cc}1/2&\sqrt3/2\\-\sqrt3/2&1/2\end{array}\right)

    I thought that b was a straightforward rotation using the rotation matrix :
    \left(\begin{array}{cc}cos\theta&sin\theta\\-sin\theta&cos\theta\end{array}\right)

    where the angle is 60 degrees, but when i graphed the unit square under this transformation it doesnt look like a simple rotation.

    Part a i am totally stuck on.
    Then graph it again! Actually, since the rotation matrix for a rotation of \theta counterclockwise is \begin{pmatrix}cos(\theta) & - sin(\theta) \\ sin(\theta) & cos(\theta)\end{pmatrix}, that is a rotation of -60 degrees, not 60 degrees.
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