This WeekÕs Homework (due Monday Jan. 10): Make sure you can do: 1,2,4,10,11,21

Turn in: Section 1.2 numbers 5,6,8,9,13, 17 as well as the following...

1.

Use induction to prove lemma 2:

$$\sum_{i = 1}^{n} c a_i = c \sum_{i =1}^n a_i.$$

2.

Prove theorem 2:

$$\sum_{i=1}^m c_{ki}\left(\sum_{j=1}^{n} b_{ij}a_{jl} \right) = \sum_{j=1}^n \left(\sum_{i=1}^{m} c_{ki} b_{ij} \right) a_{jl} .$$

3.

Given a twice differentiable real-valued function f(x) on the real line let

$$Df = \sum_{i=0}^{n}a_i \frac{d^i f}{dx^i},$$

where $\frac{d^0f}{dx^0} = f$ and the a_i are real numbers. Use induction to show for D(c f) = c D(f) for a constant c and that for a pair of such functions f₁ and f₂ that D(f₁ + f₂) = D(f₁) + D(f₂).

Proposed Topics For this Week: Reasoning and Abstraction A fair part of this course will be an effort to sharpen your proof and reasoning skill, and so we will take this on as our first task.

Our first lecture is one on reasoning and proof. In this lecture we will explore certain facts about "sequences and series" that will be useful, but more importantly in understanding the validity of these facts we will formalize certain techniques of reasoning. The interesting of which will be a technique called induction. The first three problems above will give you a chance to practice this technique.

On Friday we will explore abstraction; and in particular define and explore an abstraction known as a vector space. The remaining Homework problems concern the exploration of these vector spaces.

The X-session Topic Induction help. If you found the first lecture confusing this is a chance to discuss the homework due on Monday. No new material will be (intentionally) discussed.