K. Nicole Clark Math 311

3.2.23 - REVISED

URL: people.tamu.edu/~knc8928

3.2.23 – The linear function A: R^{2}→R^{2}
with matrix _{} is an example of a “shear
transformation.” Describe it
geometrically. (What happens to points
on a typical horizontal line, _{0 )}.

The easiest way to visualize the transformation of this function is to show the matrix multiplication written out, and see exactly what happens to x and y.

_{}

As can be seen from the multiplication above, the x
component becomes “x + ay,” and the y component remains the same. Geometrically, this means that the
transformation is along a horizontal line, y = y_{0, where}
y_{0} is a constant, or, if the original vector is a vertical line, x =
x_{0}, the line slants with a slope of 1/a. The only change in the vector is in the
x-direction; no matter what the value of x, the y value stays constant, which
means the only possible direction in which movement is possible is in the
horizontal direction.