Credit to: Jiheon Lee and Selina Phadiya for providing their lecture notes for this lecture! Thank you :)
Let us start by recalling the definition of a vector space from last time: this time we will use quantifiers to write down the vector space axioms.
Notations
In MAT223 you might be used to vectors being exclusively notated by an arrow sign over the letter:
. In more abstract linear algebra, there are so many vectors used in a single line that the arrow is often dropped. Usually, either vectors are denoted by bold symbols (like ) or, when this does not create any ambiguity, just by regular letters (like ). We will use the bold notation .
Definition
A (real) vector space is a non-empty set
with two operations: Addition (inputs two vectors, outputs a vector) Multiplication by a scalar (inputs real number (scalar) and vector, outputs vector) These operations have to satisfy the following properties (called vector space axioms):
How to understand and learn to work with this definition?
- Work with examples: learn to show why some sets with operations satisfy the requirements to be a vector space (polynomials, functions, …) and some do not.
- Prove properties of vector spaces using the formal definition.
Let us do an example of a property proof.
Statement
Given a vector space
there is only one element satisfying the property of the vector.
This is a work in progress.