Basis - Quantum Resources Site

Abstract¶

The conditions for a set of vectors to be a basis and some properties of vector basis.

Keywords:VectorsVector Spacesbasis¶

Basis Conditions¶

Let $V$ be a vector space and $a_{1}, ~a_{2}, \ldots, ~a_{n}~\in~V~\forall~n$ .

The set of vectors $\{ a_{1}, ~a_{2}, \ldots, ~a_{n} \}$ are a basis of $V$ if

$\textrm{Span}(\{ a_{1}, ~a_{2}, \ldots ~a_{n} \}) = V$
- Any vectors in $V$ can written as a linear combination of the vectors in $\{ a_{1}, ~a_{2}, \ldots ~a_{n} \}$ 💭.
The vectors $\{ a_{1}, ~a_{2}, \ldots ~a_{n} \}$ are linear independent.

Use of Basis¶

For any $\psi~\in~V$ , if $\{ a_{1}, ~a_{2}, \ldots, ~a_{n} \}$ forms a basis, there exists a unique set of scalars $\lambda_{1}, \lambda_2, \ldots, \lambda_{n}~\in~\mathbb{F}^{n}$ such that

\psi = \lambda_{1} a_{1} + \lambda_2 a_{2} + \ldots + \lambda_{n} a_{n}.

(1)

Hence, given a basis one can write any vector in the space as a sum over those vectors in the basis.

Basis Properties¶

All linear subspaces of a vector space have a basis.
All basis of a given subspace have the same number of elements.

Special Basis¶

Standard Basis

Orthonormal Basis

let $V$ be a vector space defined over $\mathbb{F}^{n}$ .

The standard basis of a vector space $V$ is given by

\begin{align*} e_{1} &= \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix}, ~ ~ e_2 &= \begin{bmatrix} 0 \\ 1 \\ \vdots \\ 0 \end{bmatrix}, ~~ e_n &= \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 1 \end{bmatrix}. \end{align*}