My Math Forum Problem with matrices.

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 June 13th, 2013, 08:08 PM #1 Newbie   Joined: Mar 2012 Posts: 21 Thanks: 0 Problem with matrices. Hi, hi! I'd like you help with the two following problems: (a) Verify that the 2 x 2 matrix $A= \begin{bmatrix} cos (\theta) &-sin (\theta) \\ sin (\theta)=&cos (\theta) \end{bmatrix}=$ is an orthogonal matrix. (b) Let T be the linear transformation with the above matrix A relative to the usual basis {i,i}. Prove that T maps each point in the plane with polar coordinates $(r, \alpha)$ onto the point with polar coordinates $(r,\alpha + \theta)$. Thus, T is a rotation of the plane about the origin, 0 being the angle of rotation. I don't have any problem with a). But I not sure how to prove the second. ¿How do I prove the relation of a transformation with its geometric interpretation?
 June 16th, 2013, 12:03 PM #2 Math Team   Joined: Sep 2007 Posts: 2,409 Thanks: 6 Re: Problem with matrices. The point with polar coordinates $(r, \alpha)$ has Cartesian coordinate $(r cos(\alpha), r sin(\alpha))$. Multiply that by the given matrix. You will need the trig identities $cos(\alpha+ \beta)= cos(\alpha)cos(\beta)- sin(\alpha)sin(\beta)$ and $sin(\alpha+ \beta)= cos(\alpha)sin(\beta)+ sin(\alpha)cos(\beta)$.

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