The linear transformation T(x1, x2) = (5x1, 15x2) has a simple geometrical meaning: stretch x1-direction 5 times and stretch x2-direction 15 times. In this exercise, we saw other examples of linear transformations with clear geometrical meanings.
In contrast, the linear transformation T(x1, x2) = (13x1 - 4x2, - 4x1 + 7x2) does not appear to have simple geometrical meaning. However, by introducing a basis v1 = (1, 2), v2 = (-2, 1), we get
T(v1) = (13×1 - 4×2, - 4×1 + 7×2) = (5, 10) = 5v1,
T(v2) = (13×(-2) - 4×1, - 4×(-2) + 7×1) = (-30, 15) = 15v2.
In other words, T stretches v1-direction and v2-direction by 5 and 15 times, respectively. From the viewpoint of a new basis, T is geometrically as simple as the last one.
For another example, the linear transformation T: R3 → R3 given by the matrix
|A = [||2/3||-1/3||-1/3||]|
appears to be messy. From this exercise, however, we have T(v1) = v1, T(v2) = v2, T(v3) = 0 for the basis v1 = (-1, 1, 0), v2 = (-1, 0, 1), v3 = (1, 1, 1). Thus T preserves (i.e., multiplying 1) in v1 and v2 directions and kills (i.e., multiplying 0) in v3 direction.
The understanding above is very useful for answering many questions about the linear transformation. For example, the powers of T(x1, x2) = (13x1 - 4x2, - 4x1 + 7x2) is characterized by
Tk(v1) = 5kv1, Tk(v2) = 15kv2,
Tk(0, 5) = Tk(2v1 + v2) = 2Tk(v1) + Tk(v2) = 2×5k(1, 2) + 15k(-2, 1) = 5k(2 - 3k2, 4 + 3k).
A direct computation would be
which is impractical to do.
Similarly, direct computation of the powers of the 3 by 3 matrix A above is almost impossible. However, the geometric meaning tells us that Ak = A for any k.
The key for the good geometrical understanding of a linear transformation is the directions for which the transformation is simply a scalar multiple. This leads to the following definition.
T(v) = λv,
then λ is an eigenvalue and v is an eigenvector.
Note that v ≠ 0 is required because the zero vector does not provide a direction. Moreover, an eigenvector is always associated with an eigenvalue. On the other hand, an engenvalue is always the scaling factor in some eigenvector direction. The two concepts are always bound together.