$ \newcommand{\bdmat}[1]{\left|\begin{array}{#1}} \newcommand{\edmat}{\end{array}\right|} \newcommand{\bmat}[1]{\left[\begin{array}{#1}} \newcommand{\emat}{\end{array}\right]} \newcommand{\coll}[2]{\bmat{r} #1 \\ #2 \emat} \newcommand{\ccoll}[2]{\bmat{c} #1 \\ #2 \emat} \newcommand{\colll}[3]{\bmat{r} #1 \\ #2 \\ #3 \emat} \newcommand{\ccolll}[3]{\bmat{c} #1 \\ #2 \\ #3 \emat} \newcommand{\collll}[4]{\bmat{r} #1 \\ #2 \\ #3 \\ #4 \emat} \newcommand{\ccollll}[4]{\bmat{c} #1 \\ #2 \\ #3 \\ #4 \emat} \newcommand{\colllll}[5]{\bmat{r} #1 \\ #2 \\ #3 \\ #4 \\ #5 \emat} \newcommand{\ccolllll}[5]{\bmat{c} #1 \\ #2 \\ #3 \\ #4 \\ #5 \emat} \newcommand{\red}[1]{{\color{red}#1}} \newcommand{\blue}[1]{{\color{blue}#1}} \newcommand{\lra}[1]{\mbox{$\xrightarrow{#1}$}} \newcommand{\rank}{\textrm{rank}} \newcommand{\row}{\textrm{row}} \newcommand{\col}{\textrm{col}} \newcommand{\null}{\textrm{null}} \newcommand{\nullity}{\textrm{nullity}} \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \renewcommand{\Arg}{\operatorname{Arg}} \renewcommand{\arg}{\operatorname{arg}} \newcommand{\adj}{\textrm{adj}} \newcommand{\mystack}[2]{\genfrac{}{}{0}{0}{#1}{#2}} \newcommand{\mystackthree}[3]{\mystack{\mystack{#1}{#2}}{#3}} \newcommand{\qimplies}{\quad\implies\quad} \newcommand{\qtext}[1]{\quad\text{#1}\quad} \newcommand{\qqtext}[1]{\qquad\text{#1}\qquad} \newcommand{\smalltext}[1]{{\small\text{#1}}} \newcommand{\svec}[1]{\,\vec{#1}} \newcommand{\querytext}[1]{\toggle{\blue{\text{?}}\vphantom{\text{#1}}}{\text{#1}}\endtoggle} \newcommand{\query}[1]{\toggle{\blue{\text{?}}\vphantom{#1}}{#1}\endtoggle} \newcommand{\smallquery}[1]{\toggle{\blue{\text{?}}}{#1}\endtoggle} \newcommand{\bv}{\mathbf{v}} \newcommand{\cyc}[2]{\cssId{#1}{\style{visibility:hidden}{#2}}} $
An applet illustrating the transformation $T_A : \R^2 \to \R^2$ for various $A.$ The black vector is the input $\vx,$ and the blue vector is the output $\color{blue} T_A(\vx) = A \vx.$
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Reflection in $x$-axis.
Reflection in $y$-axis.
Projection onto $x$-axis.
Rotation by $60^\circ$ ccw.
Rotation by $90^\circ$ ccw.
Rotate and scale.
Example 4.2.
Example 25-1.
A rank 1 example.
Custom:

(Click to move input vector. Hit 't' to toggle modes. Click on a phrase to the right to change the matrix. Enter four numbers, separated by spaces, for a custom matrix.)