# Matrices and linear transformations

• Apr 22nd 2010, 05:02 AM
ulysses123
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.
• Apr 22nd 2010, 05:21 AM
HallsofIvy
Quote:

Originally Posted by ulysses123
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?

Quote:

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.