# Basis

Let $S$ be a subset of $R_{n}$. If $B={v_{1},...,v_{k}}$ is a linearly independent set of vectors in $S$ such that $S=Span{v_{1},...,v_{k}}$, then $B$ is a basis for $S$. We define a basis for ${0}$ to be the empty set, $β$.

βBasesβ is the plural of basis.

### Bases of Subspaces

### Orthogonal and Orthonormal Bases

In linear algebra, an orthogonal basis is a set of vectors in a vector space that are mutually orthogonal to each other. More precisely, a basis is orthogonal if every pair of vectors in the basis is orthogonal, meaning their dot product is zero.

Letβs consider a vector space $V$ over a field $F$. A set of vectors $v_{1},v_{n},...,v_{n}$ in V is said to form an orthogonal basis if:

- Every vector $v_{i}$ in the set is nonzero.
- The vectors are mutually orthogonal, which means that for any distinct indices $i$ and $j$ (where $1β€i,jβ€n$), the dot product of $v_{i}$ and $v_{j}$ is zero: $v_{i}βv_{j}=0$.

### Orthonormal Bases

If each vectors in the basis has a length of 1, then the basis is called an orthonormal basis.