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.

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.