We look at some specific linear transformations from \(\R^2\) to \(R^2\) and how it transforms the unit square. Readers are encouraged to draw figures.
Example3.4.1.Exmpansion along \(x\)-axis..
\begin{equation*}
T\left(\begin{bmatrix}x\\ y \end{bmatrix} \right)= \begin{bmatrix}ax\\y \end{bmatrix} =\begin{bmatrix}a \amp 0\\0 \amp 1 \end{bmatrix} \begin{bmatrix}x\\ y \end{bmatrix}
\end{equation*}
\begin{equation*}
T\left(\begin{bmatrix}x\\ y \end{bmatrix} \right)= \begin{bmatrix}ax\\y \end{bmatrix} =\begin{bmatrix}a \amp 0\\0 \amp 1 \end{bmatrix} \begin{bmatrix}x\\ y \end{bmatrix}
\end{equation*}
where \(0\lt a\lt 1\text{.}\) Refer to the Figure 3.4.4 for \(a=1/2\text{.}\)
Figure3.4.4.\(x\)-Compression for \(a=1/2\)
Example3.4.5.Expansion along \(y\)-axis.
\begin{equation*}
T\left(\begin{bmatrix}x\\ y \end{bmatrix} \right)= \begin{bmatrix}x\\by \end{bmatrix} =\begin{bmatrix}0 \amp 0\\0 \amp b \end{bmatrix} \begin{bmatrix}x\\ y \end{bmatrix}
\end{equation*}
where \(b>0\text{.}\) Refer to Figure 3.4.6 for \(b=2\text{.}\)
Figure3.4.6.\(y\)-expansion for \(b=2\)
Example3.4.7.Compression along \(y\)-axis.
\begin{equation*}
T\left(\begin{bmatrix}x\\ y \end{bmatrix} \right)= \begin{bmatrix}x\\by \end{bmatrix} =\begin{bmatrix}0 \amp 0\\b \amp 1 \end{bmatrix} \begin{bmatrix}x\\ y \end{bmatrix}
\end{equation*}
where \(0\lt b\lt 1\text{.}\) Refer to Figure 3.4.8 for \(b=1/2\text{.}\)
Figure3.4.8.\(y\)-compression for \(b=1/2\)
Example3.4.9.Shear along positive \(x\)-axis.
\begin{equation*}
T\left(\begin{bmatrix}x\\ y \end{bmatrix} \right)= \begin{bmatrix}x+ay\\y \end{bmatrix} =\begin{bmatrix}1 \amp a\\0 \amp 1 \end{bmatrix} \begin{bmatrix}x\\ y \end{bmatrix}
\end{equation*}
where \(a>0\text{.}\) Refer to Figure 3.4.10 for \(a=1/4\text{.}\)
Figure3.4.10.Positive \(x\)-shear for \(a=1/4\)
Example3.4.11.Shear along negative \(x\)-axis.
\begin{equation*}
T\left(\begin{bmatrix}x\\ y \end{bmatrix} \right)= \begin{bmatrix}x+ay\\y \end{bmatrix} =\begin{bmatrix}1 \amp a\\0 \amp 1 \end{bmatrix} \begin{bmatrix}x\\ y \end{bmatrix}
\end{equation*}
where \(a\lt 0\text{.}\) Draw the figure for \(a=-1/4\text{.}\)
Figure3.4.12.Negative \(x\)-shear for \(a=-1/4\)
Example3.4.13.Shear along positive \(y\)-axis.
\begin{equation*}
T\left(\begin{bmatrix}x\\ y \end{bmatrix} \right)= \begin{bmatrix}x\\y+ax \end{bmatrix} =\begin{bmatrix}1 \amp 0\\a \amp 1 \end{bmatrix} \begin{bmatrix}x\\ y \end{bmatrix}
\end{equation*}
where \(a>0\text{.}\) Refer to Figure 3.4.14 for \(a=1/4\text{.}\)
Figure3.4.14.Positive \(y\)-shear for \(a=1/4\)
Example3.4.15.Shear along negative \(y\)-axis.
\begin{equation*}
T\left(\begin{bmatrix}x\\ y \end{bmatrix} \right)= \begin{bmatrix}x\\y+ax \end{bmatrix} =\begin{bmatrix}1 \amp 0\\a \amp 1 \end{bmatrix} \begin{bmatrix}x\\ y \end{bmatrix}
\end{equation*}
where \(a\lt 0\text{.}\) Refer to Figure 3.4.16 for \(a=-1/4\text{.}\)
Figure3.4.16.Negative \(y\)-shear for \(a=-1/4\)
Example3.4.17.Dilation and Contraction.
Fix a positive real number \(a\) and define \(D_a\colon \R^n\to \R^n\) by \(D_a(x)=ax\text{.}\) It is a linear map which is induced by the scalar matrix \(a I_{n}\text{.}\) ( \(D_a\) is called a dilation if \(a>0\) and a contraction if \(a\lt 1\text{.}\)) Refer to Figure 3.4.18\(D_a\colon \R^2\to \R^2\) with \(a=2\) and \(a=1/2\text{.}\)
Figure3.4.18.Dilations with \(a=1/4\) and \(a=2\text{.}\)
Example3.4.19.Streching.
Let \(a\) and \(b\) be two positive real numbers and \(A=\begin{bmatrix}a \amp 0 \\0 \amp b \end{bmatrix}\text{.}\) Define \(T\colon\R^2\to \R^2\) by \(T(x)=Ax\text{.}\) Then (i) \(T\) is stretching by a factor \(a\) along \(x\)-axis and by a factor \(b\) along \(y\)-axis if \(a,b>1\) and (ii) \(T\) is contraction by a factor \(a\) along \(x\)-axis followed by a factor \(b\) along \(y\)-axis if \(0\lt a,b\lt 1\text{.}\) Refer to Figure 3.4.20 for stretching along \(x\) by 2 and \(y\)-axes by 1.5.
Figure3.4.20.Stretching along \(x\) by 2 and \(y\)-axes by 1.5.
Example3.4.21.Geometry of linear transformation in \(\R^2\).
In this exampe, we demostrate linear transformation from \(\R^2\) to itself using the Sage interatact feature.
