Basis (Vector Space)

linear-algebra

Definition

Basis (Vector Space)

Let $V$ be a vector space. A set $B\subseteq V$ is a basis of $V$ if $B$ is linearly independent and $[B]=V$, where $[B]$ denotes the linear span of $B$.

Every $v\in V$ can be uniquely expressed as a linear combination of vectors from $B$. The coefficients of this combination are the coordinates of $v$ with respect to $B$.

Coordinate Projection

Translating coordinates between two different bases $B=\{{\mathbf{b}}_1,{\mathbf{b}}_2,\dots ,{\mathbf{b}}_n\}$ and $C=\{{\mathbf{c}}_1,{\mathbf{c}}_2,\dots ,{\mathbf{c}}_n\}$ of an $n$-dimensional vector space $V$, meaning translating coordinates ${\Phi }_B(\mathbf{x})$ into coordinates ${\Phi }_C(\mathbf{x})$, can also be solved using matrix multiplication.

Formally, the coordinate translation is performed using a linear transformation ${\Phi }_C\circ {\Phi }_B^{-1}:F^n\to F^n$, which is expressed as a matrix:

$$T_{B,C}=\left[\begin{array}{cccc}{\Phi }_C({\mathbf{b}}_1)& {\Phi }_C({\mathbf{b}}_2)& \dots & {\Phi }_C({\mathbf{b}}_n)\end{array}\right]$$

with columns $\left({\Phi }_C\circ {\Phi }_B^{-1}\right)(e_j)={\Phi }_C({\mathbf{b}}_j)$.

Thus:

$$T_{B,C}\cdot {\Phi }_B(\mathbf{x})={\Phi }_C(\mathbf{x})$$

Note that $\Phi (\mathbf{x})$ is a coordinate projection.