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(e) The images of three points lie on a line if and only if the points themselves lie on some line.
Remark It follows from parts (c), (d), and (e) that multiplication by an invertible matrix A maps triangles into triangles and
parallelograms into parallelograms.
EXAMPLE 7 Image of a Square , and is called the unit square. Sketch the image of the unit square
The square with vertices , ,
under multiplication by
Solution
Since
the image of the square is a parallelogram with vertices (0, 0), (−1, 2), (2, −1), and (1, 1) (Figure 9.2.5).
Figure 9.2.5
