A vector space is a space where vectors, matrices, polynomials, and functions live.
Geometrically, it passes through the origin (thanks to property 2 below). It could be a line, plane or hyperplane (in higher dimension).
8 full criteria to prove a vector space
Given a set
with two operations (addition and scalar multiplication), is a real-valued vector space if and it
- Is closed under both operations
- Sum of vectors is a vector in set
- Result scalar multiplication is vector in set
:
- has an additive zero vector. It acts as neutral element in a group.
- has additive inverses for all of its vectors. They act as inverse elements in a group
- keeps multiplication identity
is commutative
- both
and are associative
- both
and are distributive
Practically, we only have to prove the first three criteria (prove a subspace)
Example & non-examples
Non-examples
Specific solution to the linear system
(
is the unique solution to this system) Let is the set of solutions to this system.
: the set of geometric vectors of length 1
couldn’t become a vector space because it doesn’t have a zero vector
Type
- zero vector space
- whole space
- solution sets of homogenous system
- polynomials
- matrices