Geometry of Linear Transformations of the Plane
Let $V$ and $W$ be vector spaces. Recall that a function $T:V
\rightarrow W$ is called a linear transformation if it preserves
both vector addition and scalar multiplication:
\begin{eqnarray*}
T({\bf v_1}+ {\bf v_2}) & = & T({\bf v_1}) + T({\bf v_2}) \\
T(r{\bf v_1}) & = & rT({\bf v_1})
\end{eqnarray*}
for all ${\bf v_1, v_2} \in V$. $\qquad\qquad\qquad\qquad$
If $V = R^{2}$ and $W = R^{2}$, then $T:R^2 \rightarrow R^2$ is a
linear transformation if and only if there exists a $2 \times 2$
matrix $A$ such that $T({\bf v}) = A{\bf v}$ for all ${\bf v} \in
R^2$. Matrix $A$ is called the standard matrix for $T$. The
columns of $A$ are $T \left( \left[ {1 \atop 0} \right] \right)$ and
$T \left( \left[ {0 \atop 1} \right] \right)$, respectively. Since
each linear transformation of the plane has a unique standard matrix,
we will identify linear transformations of the plane by their standard
matrices. It can be shown that if $A$ is invertible, then the linear
transformation defined by $A$ maps parollelograms to parallelograms.
We will often illustrate the action of a linear transformation $T:R^2
\rightarrow R^2$ by looking at the image of a unit square under $T$.
Rotations
The standard matrix for the linear transformation $T:R^2 \rightarrow
R^2$ that rotates vectors by an angle $\theta$ is
$$
A = \left[\begin{array}{cc}
\cos\theta & \sin\theta \\
\sin\theta & \cos\theta
\end{array} \right].
$$
This is easily drived by noting that
\begin{eqnarray*}
T\left( \left[ {1 \atop 0} \right] \right) & = & \left[ {\cos\theta
\atop \sin\theta} \right] \\
T\left( \left[ {0 \atop 1} \right] \right) & = & \left[ {\sin\theta
\atop \cos\theta} \right].
\end{eqnarray*}


Reflections
For every line in the plane, there is a linear transformation that
reflects vectors about that line. Relection about the $x$axis is
given by the standard matrix
$$
A = \left[ \begin{array}{cc}
1 & 0\\
0 & 1
\end{array} \right]
$$
which takes the vector $\left[ {x \atop y} \right]$ to $\left[ {x
\atop y} \right]$. Reflection about the $y$axis is given by the
standard matrix
$$
A = \left[ \begin{array}{cc}
1 & 0\\
0 & 1
\end{array} \right]
$$
taking $\left[ {x \atop y} \right]$ to $\left[ {x \atop y} \right]$.
Finally, reflection about the line $y=x$ is given by
$$
A = \left[ \begin{array}{cc}
0 & 1 \\
1 & 0
\end{array} \right]
$$
and takes the vector $\left[ {x \atop y} \right]$ to $\left[ {y \atop
x} \right]$.


Expansions and Compressions
The standard matrix
$$
A = \left[ \begin{array}{cc}
k & 0 \\
0 & 1
\end{array} \right]
$$
"stretches" the vector $\left[ {x \atop y} \right]$ along the
$x$axis to $\left[ {kx \atop y} \right]$ for $k > 1$ and
"compresses" it along the $x$axis for $0~ < ~ k ~ < ~ 1$.


Similarlarly,
$$
A = \left[ \begin{array}{cc}
1 & 0 \\
0 & k
\end{array} \right]
$$
stretches or compresses vectors $\left[ {x \atop y} \right]$ to
$\left[ {x \atop ky} \right]$ along the $y$axis.


Shears
The standard matrix
$$
A = \left[ \begin{array}{cc}
1 & k \\
0 & 1
\end{array} \right]
$$
taking vectors $\left[ {x \atop y} \right]$ to $\left[ {x+ky \atop y}
\right]$ is called a shear in the $x$direction.


Similarly,
$$
A = \left[ \begin{array}{cc}
1 & 0 \\
k & 1
\end{array} \right]
$$
takes vectors $\left[ {x \atop y} \right]$ to $\left[ {x \atop y+kx}
\right]$ and is called a shear in the $y$direction.


Notes
 If finitely many linear transformations from $R^2$ to $R^2$ are
performed in succession, then there exists a single linear
transformation with thte same effect.
 If the standard matrix for a linear transformation $T: R^2
\rightarrow R^2$ is invertible,
then it can be shown that the
geometric effect of $T$ is the same as some sequence of reflections,
expansions, compressions, and shears.
In the following Exploration, you can investigate the connection
between the entries in a standard matrix and the effect the
corresponding linear transformation has geometrically.
Exploration
Key Concept
For every linear transformation $T: R^2 \rightarrow R^2$ of the plane,
there exists a standard matrix $A$ such that
$$
T({\bf v}) = A{\bf v} {\small\textrm{ for all }} {\bf v} \in R^2.
$$
Every linear transformation of the plane with an invertible
standard matrix has the geometric effect of a sequence of reflections,
expansions, compressions, and shears.
