Typical planar matrix maps include rotations, reflections, shears, scalings and projections. Use images of $e_1,e_2$ to read each transformation. Compare rotation, shear, and projection: all are linear, but only rotations preserve both lengths and orientation.
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- Matrix describing a rotation of the planeCounterclockwise rotation through angle $\theta$ has matrix $R...
- Example of a linear transformation on R^2: genericFor $T(x,y)=(2x+y,x-y)$, the standard matrix is $\begin{bmatrix}2&1\\1&-1\end{bmatrix}$. It sends $e 1$ to $(2,1)$...
- Example of a linear transformation on R^2: shearA horizontal shear is $T(x,y)=(x+ky,y)$ with matrix $\begin{bmatrix}1&k\\0&1\end{bmatrix}$. It fixes the horizontal...
- Example of a linear transformation on R^2: rotationFor a $90^\circ$ counterclockwise rotation, $R=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$. Thus $R(2,1)^T=(-1,2)^T$....
- Example of a linear transformation on R^2: projectionOrthogonal projection onto the $x$-axis is $P=\begin{bmatrix}1&0\\0&0\end{bmatrix}$, giving $P(x,y)=(x,0)$. It...
- Example of a linear transformation on R^3: rotationRotation about the $z$-axis in $\mathbb R^3$ has matrix...
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