Vector Space

A set $V$ is a vector space if there are 2 operations:

• addition
• multiplication by scalar

that satisfy the following properties:

Addition Properties

1. Closure under addition: For any vectors $\vec{u},\vec{v}\in V$, the sum $\vec{u}+\vec{v}$ is also in $V$.
2. Commutative property of addition: For any vectors $\vec{u},\vec{v}\in V$, $\vec{u}+\vec{v}=\vec{v}+\vec{u}$.
3. Associative property of addition: For any vectors $\vec{u},\vec{v},\vec{w}\in V$, $(\vec{u}+\vec{v})+\vec{w}=\vec{u}+(\vec{v}+\vec{w})$.
4. Existence of Additive Identity: There exists a vector $\vec{0}\in V$ such that for any vector $\vec{u}\in V$, $\vec{u}+\vec{0}=\vec{u}$. Also known as the zero vector.
5. Existence of Additive Inverse: For every vector $\vec{u}\in V$, there exists a vector $-\vec{u}\in V$ such that $\vec{u}+(-\vec{u})=\vec{0}$.

Scalar Multiplication Properties

1. Closure under scalar multiplication: For any scalar $c$ and vector $\vec{u}\in V$, the product $c\vec{u}$ is also in $V$.
2. Distributive Property over Vector Addition: For any scalar $c$ and vectors $\vec{u},\vec{v}\in V$, $c(\vec{u}+\vec{v})=c\vec{u}+c\vec{v}$.
3. Distributive Property over scalar addition: For any scalars $c,d$ and vector $\vec{u}\in V$, $(c+d)\vec{u}=c\vec{u}+d\vec{u}$.
4. Associative Property of scalar multiplication: For any scalars $c,d$ and vector $\vec{u}\in V$, $c(d\vec{u})=(cd)\vec{u}$.
5. Identity Element of scalar multiplication: For any vector $\vec{u}\in V$, $1\vec{u}=\vec{u}$, where $1$ is the multiplicative identity in the field of scalars.

A vector space is defined as any Set of elements that obey these properties.

Note: this means that a vector space does not have to consist of “vectors” in the traditional sense (arrows in space); it can be any set with the defined operations that satisfy the vector space axioms.