MTH 525, FALL 2006

Brief Notes for Week 6

(tentative)

Week 6:

Monday, October 2

We do examples 22 and 23 (on pages 100-103.)

We review double dual, looking briefly at Theorems 17 & 18 (which connect V with V**.)

We define hyperspace and look at Theorem 18 (hyperspaces are kernels of linear functionals.)

Do p. 105: 1, 2, 3 (on trace), 4, p. 111: 1.

Elements of L(V,W) correspond to elements of L(V*,W*) in a natural way. We draw this relationship as a “commutative” map and then walk through theorem 21 which defines the transpose of a linear transformation. This eventually leads us to the transpose matrix, via dual bases (Theorem 23, p. 113.)

Do p. 115: 1, 2, 3.

We begin chapter 4, defining a linear algebra and doing some examples, including the algebra of formal power series over a field.

We define polynomial. (And you thought you already knew the definition!)

Wednesday, October 4

Assignment 4 is collected.

Quiz 5 (on definitions) is given.

We finish chapter 4, discussing Lagrange interpolation (and its “dual” motivation).

Do p. 122: 1-3. (Also look at problem 9.) Do p. 126: 1, 3.

We define polynomial ideals. Do p. 134: 1.

We examine basic polynomial factorization properties. (Take a look at problems 5 and 6 on p. 139; these problems are the natural generalization of modular arithmetic properties of the integers.)

Last modified October 2, 2006