Question

Find the matrix A of the linear transformation T from R^2 to R^2 that rotates any vector through an angle of 30 degrees in the counterclockwise direction.

Answer 1

\[A=\left[\begin{matrix}? & ? \\ ? & ?\end{matrix}\right]\]

Answer 2

If you want to derive the \(2 \times 2\) rotation matrix by hand, think of how to transform each of the standard basis vectors.

Answer 3

\(T(e_1) = T((1, 0)) = (\cos(\theta), \sin(\theta))\)
\(T(e_2) = T((0, 1)) = (-\sin(\theta), \cos(\theta))\)

Therefore the matrix is:
\(\left(\begin{array}{cc}
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & cos(\theta)
\end{array}\right)\)

Now set \(\theta\) to 30 degrees and you have your matrix.

Answer 4

So you are sayign that my answer should be \[\left[\begin{matrix}\cos(30) & -\sin(30) \\ \sin(30) & \cos(30)\end{matrix}\right]\
because this is what I had as an answer but it says it is incorrect.

Answer 5

\[\left[\begin{matrix}\cos(30) & -\sin(30) \\ \sin(30) & \cos(30)\end{matrix}\right]\]

Answer 6

Evaluate \(\sin\) and \(\cos\) on 30 to get numeric answers.

Answer 7

\(\sin(30^\circ) = 0.5\)
\(\cos(30^\circ) = \frac{\sqrt{3}}{2}\)

Answer 8

Try numeric values and see if it accepts that.

Answer 9

It did. Thank you for your help.