MATH 3210 Exam 1 Study Guide

notevector-spaces

Question 1

We call a set $V$ a vector space over a field $\mathbb{F}$ if it has an addition and scalar multiplication operation that satisfy the following conditions:

1. Associativity of addition and scalar multiplication.
2. Commutativity of addition.
3. Existence of additive inverses.
4. Existence of an additive identity.
5. Existence of a multiplicative identity.
6. Distributivity (two ways).

Question 2

We have that for three polynomials $r(x),s(x),t(x)\in \mathcal{P}(\mathbb{R})$, $(r(x)+s(x))+t(x)=r(x)+(s(x)+t(x))$, and furthermore associativity of real scalar multiplication follows from associativity.

Question 3