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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.