Vector Space

Ingredients

A set $V$. We call its elements vectors.

A field $F$. We call its elements scalars.

A binary operation $+$ called vector addition:

$+: V \times V \rightarrow V$

A binary operation called scalar multiplication.

$F \times V \rightarrow V$

We denote scalar multiplication by juxtaposition.

We write $sv$ where $s$ is an element of $F$ and $v$ is an element of $V$.

Axioms

For the below axioms:

• $u$ and $v$ are any elements of $V$,

• $s$ and $t$ are any elements of $F$, and

• $1$ is the multiplicative identity of $F$.

Then the axioms are:

1. $V(+)$ is a commutative group.

2. $(s_1s_2)v = s_1(s_2)v$.

3. $1v = v$.

4. $s(u + v) = su + sv$.

5. $(s_1 + s_2)v = s_1v + s_2v$.

Examples

Euclidean Space

We set $\mathbb{R}$ as the field of scalars and $\mathbb{R}^n$ as the set of $n$-tuples with each element in $\mathbb{R}$.

Polynomials

We set $\mathbb{R}$ as the field of scalars and the polynomials over $\mathbb{R}$ as the vectors.

Complex numbers

Again, we can set the field of scalars to be $\mathbb{R}$ and the set of complex numbers as the set of vectors.