11/4/2008

Reminders

• Solutions in Group/Class Exercises have been posted to p:\data\math\hartlaub\linearalgebra
• Your project topic should have been approved by the end of October. If you have not chosen a topic yet, please pick one and let me know as soon as possible. You will have time to work on your project during class on Nov. 18 and Nov. 20. Your paper must be submitted by the end of the day on Nov. 20.
• Our second exam, which covers Chapters 4-6, will be next Thursday.

Section 4.3 The Matrix of a Linear Transformation

Consider a linear transformation T from V to V, where V is an n-dimensional linear space. Let B be a basis of V. Then there exists one (and only one) nxn matrix B that transforms the coordinates of f to the coordinates of T(f). The columns of B are the B-coordiante vectors of the transformation of the basic elements f1, f2, ..., fn of V. (Compare with Section 3.4 where we identified the matrix of a linear transformation for vectors v1, v2, ..., vm.)

Suppose that U and B are two bases of an n-dimensional linear space V. Then there exists one (and only one) nxn matrix S such that the [f]U = S [f]B , for all f in V. That is, the matrix S takes you from one coordinate system to another coordinate system. This invertible matrix S is called the change of basis matrix from B to U.

Let V be a linear space with two given bases U and B. Consider a linear transformation T from V to V, and let A and B be the U- and B-matrix of T, respectively. Let S be the change of basis matrix from B to U. Then A is similar to B, and AS = SB or A = SBS^(-1) or B = S^(-1)AS. (See diagram on p. 179.)

Group/Class Exercises

Section 4.3 - 1, 3, 5, 13, 21, 23, 25, 46, 49