Vector spaces

Properties

Axioms

A vector space $V$ is a collection of objects called vectors, along with two operations, addition of vectors and multiplication of a vector by a scalar from a field $\mathbb{F}$, such that the following 8 properties (aka axioms of vector spaces) hold:

Additive properties:

• Commutativity
• Associativity
• Additive identity: Existence of special vector $0$ such that $\mathbf{v}+0=\mathbf{v}$ for all $\mathbf{v}\in \mathbf{V}$. Note that this does not explicitly require $0$ to be unique, but forces that to be the case.
• Additive inverse: For every vector $\mathbf{v}\in \mathbf{V}$ there exists a vector $\mathbf{w}\in \mathbf{V}$ such that $\mathbf{v}+\mathbf{w}=0$. Note that this does not explicitly require $\mathbf{w}$ to be unique for a given $\mathbf{v}$.

Multiplicative properties:

• Multiplicative identity: $1\mathbf{v}=\mathbf{v}$ for all $\mathbf{v}\in \mathbf{V}$.
• Multiplicative associativity: $(\alpha \beta )\mathbf{v}=\alpha (\beta \mathbf{v})$ for all $\mathbf{v}\in \mathbf{V}$ and scalars alpha and beta.

Distributive properties:

• $\alpha (\mathbf{u}+\mathbf{v})=\alpha \mathbf{u}+\alpha \mathbf{v}$ for all $\mathbf{u},\mathbf{v}\in V$ and all scalars $\alpha$.
• $(\alpha +\beta )\mathbf{v}=\alpha \mathbf{v}+\beta \mathbf{v}$ for all $v\in V$ and scalars $\alpha ,\beta$.

Note

The scalars used to define vector multiplication can be drawn from any field. Usually, this is $\mathbb{R}$. Such vector spaces are called vector spaces over $\mathbb{R}$, or real vector spaces. If the scalars are complex numbers, the vector space is called a vector space over $\mathbb{C}$, or a complex vector space. Any complex vector space is a real vector space as well, since if you can multiply by a complex number, you can also multiply by a real number. If $\mathbb{F}$ is used to denote the set of scalars, then the results are true for both $\mathbb{R}$ and $\mathbb{C}$.

Important

$\varnothing$ is not a vector space, since it does not contain a $0$ vector.

Properties derivable from above axioms

• $0\in V$ is unique.
• For any $\mathbf{v}$, a vector $\mathbf{w}$ such that $\mathbf{v}+\mathbf{w}=0$ is uniquely determined. It is denoted by $-\mathbf{v}$.
• $0\mathbf{v}=0$.
• $(-1)\mathbf{v}=-\mathbf{v}$.

${\mathbb{R}}^n$

${\mathbb{R}}^n$ is a vector space.

$${\mathbb{R}}^n=\left\{\begin{array}{c}\left(\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right)|x_i\in \mathbb{R}\end{array}\right\}$$

Note

It can be shown that all vector spaces are isomorphic with ${\mathbb{R}}^n$.

${\mathbb{R}}^n$ consists of all columns of size $n$. Addition and multiplication are defined entry-wise.

$$\alpha \left(\begin{array}{c}{v}_1\\ v_2\\ ⋮\\ v_n\end{array}\right)=\left(\begin{array}{c}\alpha v_1\\ \alpha v_2\\ ⋮\\ \alpha v_n\end{array}\right),\left(\begin{array}{c}{v}_1\\ v_2\\ ⋮\\ v_n\end{array}\right)+\left(\begin{array}{c}{w}_1\\ w_2\\ ⋮\\ w_n\end{array}\right)=\left(\begin{array}{c}{v}_1+w_1\\ v_2+w_2\\ ⋮\\ v_n+w_n\end{array}\right)$$

K-cell

If $a_i<b_i$ for $i=1,\dots ,k$, the set of all points $\mathbf{x}=(x_1,x_2,\dots ,x_k)$ in ${\mathbb{R}}^k$ whose coordinates satisfy the inequalities $a_i\le x_i\le b_i$ for $1\le i\le k$ is called a k-cell. A 1-cell is an interval, a 2-cell is a rectangle, etc.

${\mathbb{P}}_n$

The space ${\mathbb{P}}^n$ consists of all polynomials of degree up to $n$.