Matrix multiplication/Rotations

[Jerbearrrrrr]

An engineer came to me with the following problem

Homework Statement

Suppose
$y = A x$
where $x,y \in R^2$
and $A = \left( \begin{array}{cc} \cos \theta & -\sin \theta\\ \sin \theta & \cos \theta \end{array} \right)$.
Show that y is an anticlockwise rotation of x about the origin.

Homework Equations

None.
Maybe definition of SO(2,R).

The Attempt at a Solution

$\square$

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I kind of don't understand the question.
How can you prove a definition?
Is this question not asking something like, "prove that average speed equals distance over time"?
I guess they want the student to draw a load of triangles? Or perhaps express x and y in terms of polar coordinates to make it more obvious that it's a rotation? Or maybe to show that it can be written as a product of two reflections? Or show that |x| = |y|...but that doesn't make any comments about the angle.
Any suggestions?
Thanks

 

[praharmitra]

i think what is being asked of you is to forget about SO(2) and everything. Just draw two point vectors in a 2-d plane separated by an angular distance of \theta.

Now, by just applying your knowledge of geometry, show that the new coordinates of y in terms of the old coordinates x, is the exact same equations as written above in matrix form.

 

[HallsofIvy]

Suppose you were to apply that matrix to (1, 0). What would the result be? Suppose you were to apply it to (0, 1)? How do those points relate to (1, 0) and (0, 1)?

 

[Jerbearrrrrr]

Thanks. I thought they were after something like that. Expand sin(A+B) and wave some hands.

Kinda just looking for a definition of 'rotation' in the context of "mathematics for engineers" I suppose, since my friend (whom I'm posting on behalf of) couldn't supply one from lecture notes.

To prove a Foo is a Bar, you have to know what Foos and Bars actually are. And I suppose since the only geometry I've done has been in bits of algebra courses, I would naturally think of group-context definitions xD