What does a linear transformation do to a vector in $\mathbb{R}^2$

Question

$\newcommand{\Reals}{\mathbb{R}}$Doing some linear algebra. This question has me intrigued. I imagine it causes the vector to rotates the $x$ axis. But I mean, is there more to it? $$ T_{A(\theta)}: \Reals^2 \to \Reals^2,\qquad A(\theta) = \left[\begin{array} {rr}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{array}\right]. $$ What, geometrically, does the linear transformation $T_{A(\theta)}$ do to a vector in $\Reals^2$?

Answer 1

Take the standard basis vector of $\mathbb{R}^2$ which is $(1, 0)$ and $(0, 1)$ and then draw it on the $X$ and $Y$ axis as an arrow vector. Now find the image of $A_\theta (1, 0)$ and $A_\theta (0, 1)$ for different values of $\theta$ and plot them again. You may see how $A_\theta$ rotates these vectors by an angle $\theta$.

To get further insight draw a rectangle in $\mathbb{R}^2$ with vertices $(0, 0)$, $(1, 0)$ $(1, 1)$ $(0, 1)$. Now find image of each of these vectors by assuming a particular values of $\theta$. Now analyze how the rectangle is rotated by an angle $\theta$ by rotation matrix $A_{\theta}$.

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