MAT223 Lecture 15

• $T\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}yx+4\\ x+4y\end{array}\right]$
• $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$
  • Scale by a factor of $C$
    • $\left[\begin{array}{cc}c& 0\\ 0& c\end{array}\right]$
    • $\left[\begin{array}{cc}{\overrightarrow{e}}_1& {\overrightarrow{e}}_2\end{array}\right]$
    • ${\overrightarrow{e}}_n$ are basically your $\overrightarrow{i},\overrightarrow{j},\overrightarrow{k}$
  • Stretch in $x$ direction by factor of $a$
    • We need $2{\overrightarrow{e}}_1$
    • $\left[\begin{array}{cc}a& 0\\ 0& 1\end{array}\right]$
  • Stretch in $y$ direction by factor of $b$
    • We need $2{\overrightarrow{e}}_2$ if we scale by 2
    • $\left[\begin{array}{cc}1& 0\\ 0& b\end{array}\right]$
  • $T(\overrightarrow{x})=A_t\overrightarrow{x}$
    • $A_t=\left[\begin{array}{cccc}T({\overrightarrow{e}}_1)& T({\overrightarrow{e}}_2)& \dots & T({\overrightarrow{e}}_n)\end{array}\right]$
  • Rotation counter-clockwise by $\theta$
    • $\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$
    • This is essentially because we have rotating vectors along the unit circle.
      • #tk can I use this in my model of cylinder pressures based on the angle of the crankshaft?
    • Example:
      • Rotation by $\frac{\pi }{2}$
      • $T({\overrightarrow{e}}_1)=\left[\begin{array}{c}0\\ 1\end{array}\right]$
      • $T({\overrightarrow{e}}_2)=\left[\begin{array}{c}-1\\ 0\end{array}\right]$
  • Reflection across $y=mx$
    • $m=0⟹y=0x⟹y=0$
      • This is a reflection across the $x$-axis.
      • $T({\overrightarrow{e}}_1)=\left[\begin{array}{c}1\\ 0\end{array}\right]$
      • $T({\overrightarrow{e}}_2)=\left[\begin{array}{c}0\\ -1\end{array}\right]$
    • $\frac{1}{m^2+1}\left[\begin{array}{cc}1-m^2& 2m\\ 2m& m^2-1\end{array}\right]$
  • Projection onto $y=mx$
    • $m=0⟹y=0x⟹y=0$
      • This is a projection onto the $x$-axis.
      • $T({\overrightarrow{e}}_1)=\left[\begin{array}{c}1\\ 0\end{array}\right]$
      • $T({\overrightarrow{e}}_2)=\left[\begin{array}{c}0\\ 0\end{array}\right]$
      • We flatten the $y$ vals down to $0$
    • $\frac{1}{m^2+1}\left[\begin{array}{cc}1& m\\ m& m^2\end{array}\right]$
  • #tk try to derive the $m^2\dots$ formulas for reflection and projection by checking $m=0,\pm 1$
• Poll
  • $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$ is not linear because:
    • $T\left[\begin{array}{c}0\\ 0\end{array}\right]=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$
      • ${\mathbb{R}}^2\to {\mathbb{R}}^3$
      • Can $0$ map to something?
      • That would mean that $T$ is not linear because $T(\overrightarrow{0})$ should equal $\overrightarrow{0}$.
    • $T\left[\begin{array}{c}1\\ 0\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$
      • ${\mathbb{R}}^2\to {\mathbb{R}}^3$
      • This could be possible, like the first entry is just $0x$
    • $T\left[\begin{array}{c}1\\ 1\end{array}\right]\ne T\left[\begin{array}{c}1\\ 0\end{array}\right]+T\left[\begin{array}{c}0\\ 1\end{array}\right]$
      • $1=1+0$
      • $1=0+1$
      • If this is true, then it's not linear.
      • Because to be linear we need $T(\overrightarrow{a}+\overrightarrow{b})=T(\overrightarrow{a})+T(\overrightarrow{b})$
• Activity
  • Describe the transformations
  • $T\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}2x\\ 3y\end{array}\right]$
    • Stretch in $x$ direction by factor of $2$
    • Stretch in $y$ direction by factor of $3$
    • b
  • $Q\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}3x+3y\\ x-y\end{array}\right]$
    • $A_t\overrightarrow{x}$
    • $A_t\left[\begin{array}{c}x\\ y\end{array}\right]$
    • $\left[\begin{array}{cc}3& 3\\ 1& -1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$
    • $T({\overrightarrow{e}}_1)=\left[\begin{array}{c}3\\ 1\end{array}\right]$
    • $T({\overrightarrow{e}}_2)=\left[\begin{array}{c}3\\ -1\end{array}\right]$
    • G
  • $R$ is a reflection across the $x$ axis:
    • This means that $(5,10)\to (5,-10)$
    • So $R\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}x\\ -y\end{array}\right]$
    • d
  • $P$ is the projection onto the $y$ axis:
    • $(x,y)\to (0,y)$
    • $P\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}0\\ y\end{array}\right]$
    • h
  • $S$ is the counter clockwise rotation by $\frac{\pi }{2}$
    • $\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$
    • a
  • $U=R\circ S$
    • First rotate, then reflect across the $x$ axis.
    • e
  • Paul Piunno: paul.piunno@utoronto.ca
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