Basis

Given a Vector space VV, a basis is a set of linearly independent vectors {e1,…,en}\{ \mathbf{e}_{1},\ldots,\mathbf{e}_{n} \} such that for each vector v\mathbf{v} in VV, there exist one and only one set of constants {α1,…,αn}∈Rn\{ \alpha_{1},\ldots,\alpha_{n} \}\in \mathbb{R}^{n} for which the Linear combination of basis vectors using these values is exactly v\mathbf{v}. In symbols:

{e1,…,en}∈V such that ∀v∈V, ∃! {α1,…,αn} such that v=∑i=1nαiei\{ \mathbf{e}_{1},\ldots,\mathbf{e}_{n} \}\in V\text{ such that }\forall \mathbf{v}\in V,\ \exists!\ \{ \alpha_{1},\ldots,\alpha_{n} \}\text{ such that }\mathbf{v}=\sum_{i=1}^{n} \alpha_{i}\mathbf{e}_{i}

The numbers αi\alpha_{i} are said to be the components of v\mathbf{v} with respect to this basis.

All vectors in VV can be expressed as a linear combination of the basis vectors, given a suitable set of values. The Scalar product between a vector v\mathbf{v} and the ii-th basis vector ei\mathbf{e}_{i} gives the ii-th component of v\mathbf{v}:

v⋅ei=(∑j=1nαjej)⋅ei=∑j=1nαj(ej⋅ei)=∑j=1nαjδij=αi\mathbf{v}\cdot \mathbf{e}_{i}=\left( \sum_{j=1}^{n} \alpha_{j}\mathbf{e}_{j} \right)\cdot \mathbf{e}_{i}=\sum_{j=1}^{n} \alpha_{j}(\mathbf{e}_{j}\cdot \mathbf{e}_{i})=\sum_{j=1}^{n} \alpha_{j}\delta_{ij}=\alpha_{i}

Vectors of a basis are always orthogonal to each other. If they are also normalized, the basis is said to be orthonormal. In an orthonormal basis, ei⋅ej=δij\mathbf{e}_{i}\cdot \mathbf{e}_{j}=\delta_{ij}, using the Kronecker delta.

Connection to physics

The choice of a basis corresponds to the choice of a coordinate system, the most common of which are Cartesian coordinates, polar coordinates, cylindrical coordinates and spherical coordinates.