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.