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 \(\mathbf{u} \in U\), \(L(\alpha \mathbf{u}) = \alpha L(\mathbf{u})\)

Additional operations

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.