MTH 525, FALL 2006

Brief Notes for Week 4

Week 4:

Monday, September 18

Assignment 2 is collected.

We begin Chapter 3, Linear Transformations (section 3.1), defining linear transformations, putting them in the larger context of mathematical “morphisms”, and giving a variety of examples.

Major results from this lecture:

A linear transformation is uniquely defined on its basis.

Nullity + Rank = Dimension of the codomain.

Rank = Row rank.

Do the following problems: 73: 1-7.

For Wednesday’s class, read section 3.2.

Wednesday, September 20

We do sections 3.2, 3.3 and begin section 3.4.

In section 3.2 we look at the algebra of linear transformations. We find a “good” basis for the vector space L(V,W).

We examine L(V,V), the space of linear operators on V.

We look at composition of linear transformations and show that it is equivalent to matrix multiplication! (see p. 78 and then section 3.4.)

We define nonsingular linear transformations.

Theorem 8: A linear transformation is nonsingular iff it maps independent sets to independent sets.

We glance at Theorem 9 (equivalent statements about “nonsingular”.)

Do 83: 1, 3, 4.

We look at isomorphisms (section 3.3, very short.)

We begin section 3.4, representing linear transformations as matrices (under the assumption that the underlying vector spaces are finite dimensional.)

This “change of basis” matrix P has columns which are the coordinates of basis #2 written with respect to basis #1 (see p. 91, in chapter 3.)

Last modified October 1, 2006