Vector space

Intuition

A vector space is a set containing vectors, whose essential properties are that they can be added together and multiplied by scalars. Vectors can be thought of as “arrows,” as they are usually introduced in earlier classes, but one later realizes that other objects can also be regarded as vectors, such as matrices, polynomials, functions, etc.

The addition of vectors and the addition of functions illustrate the same algebraic structure.

The formal definition below ensures that these two operations, addition and scalar multiplication, behave consistently with the usual rules of algebra.

Formal definition

A vector space over a field $\mathbb{K}$ is a set $E$ equipped with two operations :

• Vector addition $+:E\times E \rightarrow E$
• Scalar multiplication $\cdot:\mathbb{K}\times E \rightarrow E$

such that the following properties hold for all $u,v,w\in E$ and all $\lambda,\mu\in K$:

1. $E$ is an abelian group with $+$.
2. Distributivity over vector addition: $\lambda\cdot(u+v)=\lambda \cdot u+\lambda \cdot v$
3. Distributivity over scalar addition: $(\lambda+\mu)\cdot u=\lambda\cdot u+\mu\cdot u$
4. Compatibility with field multiplication: $\lambda\cdot(\mu\cdot u)=(\lambda \mu)u$
5. Identity scalar: $1\cdot u=u$ where $1$ is the multiplicative identity of $\mathbb{K}$.