Basis

Let $S$ be a subset of ${\mathbb{R}}^n$. If $B=\{{\vec{v}}_1,...,{\vec{v}}_k\}$ is a linearly independent set of vectors in $S$ such that $S=Span\{{\vec{v}}_1,...,{\vec{v}}_k\}$, then $B$ is a basis for $S$. We define a basis for $\{\vec{0}\}$ to be the empty set, $\varnothing$.

“Bases” is the plural of basis.

A basis is a set of vectors such that their linear combination can generate any vector in the entire vector space.

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.

Let’s 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:

1. Every vector $v_i$ in the set is nonzero.
2. The vectors are mutually orthogonal, which means that for any distinct indices $i$ and $j$ (where $1\le i,j\le n$), the dot product of $v_i$ and $v_j$ is zero: $v_i\cdot v_j=0$.

Orthonormal Bases

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

Orthonormal = orthogonal + normalized