Lecture notes from lecture #2 of [[MATH 236]]. In this lecture, we got a brief introduction to complex numbers, and were given the definition of a vector space (along with some examples).
[TOC]
Note that
We can then define the operations of addition and multiplication in the standard way.
Exercise: Prove that
Definition: A set
- Commutativity of addition
- Associativity of addition
- Existence of an additive identity
- Existence of additive inverses
- Existence of a multiplicative identity (in
$\mathbb F$ - left and right multiplication) - Distributivity:
$a(u+v) = au +av$ for$a \in \mathbb F$ and$u, v \in V$ , as well as$(a+b)u = au + bu$ for$a, b \in \mathbb F$ and$u \in V$ .
-
$\mathbb R^2$ . An element of this vector space is, well, a vector, starting at the origin. To prove that this is a vector space, we define addition and scalar multiplication in the usual way, then verify the properties. -
$\mathbb F^n$ in general, over$\mathbb F$ . - Polynomials.
$x \mapsto a_0 + a_1x + a_2x^2 + \ldots + a_nx^n$ . - Matrices. $M_{m\times n}(\mathbb F) = $ the set of
$m\ times n$ matrices with entries in$\mathbb F$ . We define addition only on matrices with the same dimensions.
Exercises: Verify that the following are vector spaces: