I'm new to Linear Algebra and I'm having a hard time wrapping my head around linear transformations, specifically a rotation.
From Anton's book (Elementary Linear Algebra, 11th Edition) he states:
$T(e_1) = T(1,0) = (\cos\theta, \sin\theta)$
and
$T(e_2) = T(0,1) = (-\sin\theta, \cos\theta)$
and
$A = [T(e_1) | T(e_2)] = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$
When I rotate a vector $\begin{bmatrix} x\\y \end{bmatrix}$ I get
$\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} x \cdot\cos\theta \, - \, y \cdot\sin\theta \\ x \cdot\sin\theta \, + \, y \cdot\cos\theta \end{bmatrix}$
Correct me if I'm wrong, but I thought that column 1 of $A$ $\begin{bmatrix} \cos\theta \\ \sin\theta \end{bmatrix}$, holds the 'x' values and column 2 holds the 'y' values. What I'm confused about is why does $x'$ contain both an $x$ component and a $y$ component?
When I transform $\begin{bmatrix}1 \\ 0 \end{bmatrix}$ by a rotation $\theta$, the new $x$ value is just $\cos\theta$ while the new $y$ value is just $\sin\theta$. I don't understand why with the matrix transformation, $x'$ and $y'$ get both $x$ and $y$ components summed together.
I hope I'm making sense.