Math 2406: Abstract Vector
Spaces
Lecture 14 Plan
(Thursday, October 11, 2007).
Apostol 4.10-4.11.
We will go over the abstract of idea of a matrix representation of a
linear transformation. Any matrix has to be defined together with bases in the domain and
range spaces. The most common bases to use for an m-by-n
matrix which maps from R^n to R^m are the standard coordinate
vectors. But it is useful to recognize that we can define
matrices with respect to other bases. This helps us understand better
the fact that a linear transformation can be understood completely by
how it carries any basis into any other basis. It can also useful to
think of matrices for different function bases, such as polynomials,
exponential and trigonometric function.
The diagonal-matrix theorem 4.14 relates the matrix representation to
our concepts of range, null space, rank and nullity.
Review:
1. How
can we express the identity transformation from R^2 to R^2 using the
2-by-2 matrix [2 0; 0 1]? (hint: what basis would this matrix be
defined with respect to?)
2.
How many zeros does the matrix in Theorem 4.14 have on the diagonal? How does this number relate to the range or null space?
Homework 6
due at 5 p.m.
Thursday, October 18, 2007.
1. Apostol 4.8 Exercise 12.
2. Apostol 4.8 Exercise 20.
3. Apostol 4.8 Exercise 23.
4. Apostol 4.8 Exercise 27.
5. Apostol 4.12 Exercise 5.
6. Apostol 4.12 Exercise 10.
7. Apostol 4.12 Exercise 16.