K. Nicole Clark                                                                                                       Math 311

2/9/2004                                                                                                                  Fulling

3.2.23 - REVISED

URL: people.tamu.edu/~knc8928

3.2.23 – The linear function A: R2→R2 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 = y0,  where y0 is a constant, or, if the original vector is a vertical line, x = x0, 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.