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

Formally a vector space $$V$$ is a set with two operations $$+$$ and $$*$$ that satisfy the following properties

• if $$\mathbf{u},\mathbf{v} \in V$$ then $$\mathbf{u + v} \in V$$
• $$\mathbf{u + v} = \mathbf{v + u}$$
• $$\mathbf{u + (v + w)} = \mathbf{(u + v) + w}$$
• There is an special element called the zero vector $$\mathbf{0} \in V$$ such that $$\mathbf{u + 0} = \mathbf{0 + u} = \mathbf{u}$$
• For every $$\mathbf{u} \in V$$ there's an inverse element $$-\mathbf{u}$$ such that $$\mathbf{u + (-u)} = \mathbf{0}$$
• 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 vectors spaces

• Segments on the plane and 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 vectors space $$V$$ is a subspace if

• For all $$\mathbf{u,v} \in U$$, $$\mathbf{u+v} \in U$$
• 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 $$\mathbf{b_1, b_2, \ldots, b_n}$$ is a basis then any linear combination of the basis will have the form

$\mathbf{v} = a_1 \mathbf{b_1} + a_2 \mathbf{b_2} + \ldots + a_n \mathbf{b_n}$

The numbers $$a_1, a_2, \ldots, a_n$$ 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 vectors spaces is linear if it preserves addition and multiplication with scalars as defined above, formally a map $$L: U \rightarrow V$$ is linear if

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

### Norm

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

• $$\norm{\mathbf{v}} \geq 0$$, $$\norm{\mathbf{v}} = 0$$ only if $$\mathbf{v} = \mathbf{0}$$
• $$\norm{\alpha \mathbf{v}} = \alpha \norm{\mathbf{v}}$$
• $$\norm{\mathbf{v_1} + \mathbf{v_2}} \leq \norm{\mathbf{v_1}} + \norm{\mathbf{v_2}}$$ (triangle sides)

### Scalar product

The scalar product of two vectors is a function $$f: V \times V \rightarrow \mathbb{R}$$, the function is commonly denoted as $$\left \langle \mathbf{v_1}, \mathbf{v_2} \right \rangle$$ and satisfies

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

References
• Nearing, J. (2016). Vector spaces. [online] Physics.miami.edu. Available at: http://www.physics.miami.edu/~nearing/mathmethods/vector_spaces.pdf [Accessed 15 Mar. 2016].
• Bærentzen, J. A., Gravesen, J., Anton François, & Aanæs, H. (2012). Guide to computational geometry processing: foundations, algorithms, and methods. London: Springer.