Linear Transformation involving pi/2

[dwn]

Resource: Linear Algebra (4th Edition) -David C. Lay

I understand that there are identities associated with transformations, but what I don't understand is when the transformation is rotated about the origin through an angle β. I believe β in this case is $\frac{}{}\pi/2$

$\left[1,0\right]$ into [cos($\beta$) , sin($\beta$)]
$\left[0,1\right]$ into [-sin($\beta$), cos($\beta$)]

Can someone please explain to me why this is the case? Why do these values suddenly translate to trig identities?

Thanks!

 

[Fredrik]

I don't understand the question. "...β in this case..." What case?

Are you asking why (1,0) rotated counterclockwise by an angle of β is (cos β,sin β)? This is a very common way to define sin and cos. Do you want to use another definition?

 

[dwn]

I suppose my questions reiterates my confusion...haha.

There is something I'm not grasping in the definition of this counterclockwise rotation. How am I suppose to know the positive/negative values of the matrix and whether they're sin or cosine..? What if this type of rotation is not $\pi$/2?

 

[Fredrik]

\begin{pmatrix}\cos\beta & -\sin\beta\\ \sin\beta & \cos\beta\end{pmatrix} is the matrix representation of a counterclockwise rotation by an arbitrary angle β. When $\beta=\pi/2$, we have $\sin\beta=1$ and $\cos\beta=0$, so the matrix representation of a counterclockwise rotation by $\pi/2$ is
\begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}

 

[dwn]

Sometimes its difficult "to see the wood for the forest". That's all I will say.

Thanks for your clarifying this point for me.