A vector space is a set whose elements are called “vectors” (denoted as $\overrightarrow{v}$ or $\mathbf{v}$), which have two operations defined on them: addition of vectors and multiplication of a scalar by a vector.

Formally, a vector space $V$ is a set with two operations $+$ and $\ast$ that satisfy the following properties:

• If $\mathbf{u},\mathbf{v}\in V$, then $\mathbf{u}\mathbf{+}\mathbf{v}\in V$.
  • $\mathbf{u}\mathbf{+}\mathbf{v}=\mathbf{v}\mathbf{+}\mathbf{u}$
  • $\mathbf{u}\mathbf{+}\mathbf{(}\mathbf{v}\mathbf{+}\mathbf{w}\mathbf{)}=\mathbf{(}\mathbf{u}\mathbf{+}\mathbf{v}\mathbf{)}\mathbf{+}\mathbf{w}$
  • There is a special element called the zero vector $\mathbf{0}\in V$ such that $\mathbf{u}\mathbf{+}\mathbf{0}=\mathbf{0}\mathbf{+}\mathbf{u}=\mathbf{u}$.
  • For every $\mathbf{u}\in V$, there’s an inverse element $-\mathbf{u}$ such that $\mathbf{u}\mathbf{+}\mathbf{(}\mathbf{-}\mathbf{u}\mathbf{)}=\mathbf{0}$.
• If $\mathbf{u}\in V$ and $\alpha \in \mathbb{R}$, then $\alpha \mathbf{u}\in V$.
  • $(\alpha +\beta )\mathbf{u}=\alpha \mathbf{u}+\beta \mathbf{u}$
  • $\alpha (\beta \mathbf{u})=(\alpha \beta )\mathbf{u}$
  • $1\cdot \mathbf{u}=\mathbf{u}$

Notable examples of vector spaces:

• Segments on the plane and in space; addition uses the parallelogram law, and multiplication by a scalar scales the segment.
• The set of $n\times n$ matrices, with addition defined by element.
• The set of all polynomials.
• The space consisting of the zero vector alone: $\mathbf{0}$.

Vector subspaces

A subset $U\subseteq V$ of a vector space $V$ is a subspace if:

• For all $\mathbf{u}\mathbf{,}\mathbf{v}\in U$, $\mathbf{u}\mathbf{+}\mathbf{v}\in U$.
• For all $\alpha \in \mathbb{R}$ and $\mathbf{u}\in U$, $\alpha \mathbf{u}\in U$.

Linear dependence

A set of vectors is linearly dependent if one element from the set can be written as a linear combination of the other elements in the set. If this cannot be done, then the set is linearly independent, which is also known as a basis for some vector space. The dimension is the number of elements in the basis. If ${\mathbf{b}}_{\mathbf{1}}\mathbf{,}{\mathbf{b}}_{\mathbf{2}}\mathbf{,}\dots \mathbf{,}{\mathbf{b}}_{\mathbf{n}}$ is a basis, then any linear combination of the basis will have the form:

The numbers $a_1,a_2,\dots ,a_n$ are called the components of $\mathbf{v}$ in the specified basis. Note that the basis doesn’t need to be orthogonal nor have unit vectors.

The set of vectors $[1,0,0],[0,1,0],[0,0,1]$ is an example of a basis of dimension 3

Linear maps

A map between vector spaces is linear if it preserves addition and multiplication with scalars as defined above. Formally, a map $L:U\to V$ is linear if:

• For all $\mathbf{u}\mathbf{,}\mathbf{v}\in U$, $L(\mathbf{u}\mathbf{,}\mathbf{v})=L(\mathbf{u})+L(\mathbf{v})$.
• For all $\alpha \in \mathbb{R}$ and $\mathbf{u}\in U$, $L(\alpha \mathbf{u})=\alpha L(\mathbf{u})$.

Additional operations

Norm

The norm of a vector is denoted by $|\mathbf{v}|$ and satisfies:

• $|\mathbf{v}|\ge 0$; $|\mathbf{v}|=0$ only if $\mathbf{v}=\mathbf{0}$.
• $|\alpha \mathbf{v}|=\alpha |\mathbf{v}|$.
• $|{\mathbf{v}}_{\mathbf{1}}+{\mathbf{v}}_{\mathbf{2}}|\le |{\mathbf{v}}_{\mathbf{1}}|+|{\mathbf{v}}_{\mathbf{2}}|$ (triangle inequality).

Scalar product

The scalar product of two vectors is a function $f:V\times V\to \mathbb{R}$. The function is commonly denoted as $⟨{\mathbf{v}}_{\mathbf{1}},{\mathbf{v}}_{\mathbf{2}}⟩$ and satisfies:

• $⟨\mathbf{w}\mathbf{,}\mathbf{(}\mathbf{u}\mathbf{+}\mathbf{v}\mathbf{)}⟩=⟨\mathbf{w}\mathbf{,}\mathbf{u}⟩+⟨\mathbf{w}\mathbf{,}\mathbf{v}⟩$.
• $⟨\mathbf{w},\alpha \mathbf{v}⟩=\alpha ⟨\mathbf{w}\mathbf{,}\mathbf{v}⟩$.
• $\left \langle \mathbf{v,v} \right \rangle \geq 0.