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Figure 9.2.3
A shear in the y-direction with factor k is a transformation that moves each point parallel to the y-axis by an amount to
the new position . Under such a transformation, points on the y-axis remain fixed, and points farther from the y-axis
move a greater distance than those that are closer.
It can be shown that shears are linear transformations. If is a shear with factor k in the x-direction, then
so the standard matrix for T is
Similarly, the standard matrix for a shear in the y-direction with factor k is
Remark Multiplication by the identity matrix is the identity operator on . This operator can be viewed as a rotation
through 0°, or as a shear along either axis with , or as a compression or expansion along either axis with factor .
