# Vector Signals and Invariant Systems: Re-Tooling Linear Systems Theory
## 3 Notation and Terminology for Geometric Algebra  
Although Clifford algebras may be defined over general fields, in GA, only real numbers are associated with scalar quantities, which are denoted using lowercase italic font, i.e., \(a,k,\alpha,\theta,t,\tau\in\mathbb{R}\). Traditional (polar) vectors are associated with directed lengths and denoted using lowercase upright bold font, i.e., \(\mathbf{u},\mathbf{u}^{\prime},\mathbf{e}_{1},\mathbf{x},\mathbf{y},\mathbf{ h}\in\mathbb{L}^{n}\), where \(\mathbb{L}^{n}\) denotes the space of n-dimensional directed lengths. Traditional axial vectors are closely related with directed areas (bivectors) and denoted using uppercase upright bold font, i.e., \(\mathbf{B},\in\mathbb{B}^{n}\) where \(\mathbb{B}^{n}\) denotes the space of n-dimensional directed areas. The GA is a _vector space_ with an additional product--the _geometric product_, which unifies the inner (dot) and exterior/outer (wedge) product between two vectors \(\mathbf{u},\mathbf{v}\) into a single geometric product  
\[\mathbf{u}\mathbf{v}=\mathbf{u}\cdot\mathbf{v}+\mathbf{u}\wedge\mathbf{v}. \tag{15}\]  
A summary of the properties of the dot, wedge, and geometric products is provided in Table 1.  
For Euclidean spaces, the vector space \(\mathbb{L}^{n}\) can be extended to the geometric algebra \(\mathbb{G}^{n}\). The canonical basis for \(\mathbb{G}^{n}\) is determined by an orthonormal basis for \(\mathbb{L}^{n}\) where the orthonormal basis vectors satisfy  
\[\mathbf{e}_{i}\mathbf{e}_{j}= \begin{cases}1,&i=j\\ -\mathbf{e}_{j}\mathbf{e}_{i},&i\
eq j\end{cases}  \tag{16}\]  
where \(i,j=1,2,\ldots,N\). The elements of \(\mathbb{G}^{n}\) are called multivectors and denoted by an uppercase italic font, i.e., \(X,R\in\mathbb{G}^{n}\). One exception to the notational rules provided above is that simple unit bivectors \(\mathbf{j}\in\{\mathbf{B}\in\mathbb{B}^{n}:\|\mathbf{B}\|=1\}\) are denoted with _lowercase_ upright bold font \(\mathbf{j}\) because they have the property  
\[\mathbf{j}^{2}=-1. \tag{17}\]  
Furthermore, they play a role similar to that of the traditional imaginary number (as a generator of rotations). In a similar manner to how unit vectors are used to represent \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Dot Product** & **Wedge Product** & **Geometric Product** \\ \hline
**Commutative** & \(\mathbf{u}\cdot\mathbf{v}=\mathbf{v}\cdot\mathbf{u}\) & \(\mathbf{u}\wedge\mathbf{v}=-\mathbf{v}\wedge\mathbf{u}\) & \(\mathbf{u}\mathbf{v}\
eq\mathbf{v}\mathbf{u}\) \\
**Associative** & \((\mathbf{u}\cdot\mathbf{v})\cdot\mathbf{w}\
eq\mathbf{u}\cdot(\mathbf{v} \cdot\mathbf{w})\) & \((\mathbf{u}\wedge\mathbf{v})\wedge\mathbf{w}=\mathbf{u}\wedge(\mathbf{v} \wedge\mathbf{w})\) & \((\mathbf{u}\mathbf{v})\mathbf{w}=\mathbf{u}(\mathbf{v}\mathbf{w})\) \\
**Parallel** & \(\mathbf{u}\cdot\mathbf{v}\
eq 0\) & \(\mathbf{u}\wedge\mathbf{v}=0\) & \(\mathbf{u}\mathbf{v}=\mathbf{u}\cdot\mathbf{v}\) \\
**Perpendicular** & \(\mathbf{u}\cdot\mathbf{v}=0\) & \(\mathbf{u}\wedge\mathbf{v}\
eq 0\) & \(\mathbf{u}\mathbf{v}=\mathbf{u}\wedge\mathbf{v}\) \\
**Distributive** & \((\mathbf{u}+\mathbf{v})\cdot\mathbf{w}=\mathbf{u}\cdot\mathbf{v}+\mathbf{v} \cdot\mathbf{w}\) & \((\mathbf{u}+\mathbf{v})\wedge\mathbf{w}=\mathbf{u}\wedge\mathbf{v}+\mathbf{v} \wedge\mathbf{w}\) & \((\mathbf{u}+\mathbf{v})\mathbf{w}=\mathbf{u}\mathbf{v}+\mathbf{v}\mathbf{w}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties associated with the dot product, wedge product, and geometric product. lines through the origin (where there are two unit vectors \(\pm\mathbf{v}\) corresponding to every line), unit bivectors may be used to represent planes through the origin. Thus, we will use the language \"the plane \(\mathbf{j}\)\" as an abbreviation for \"the plane through the origin whose unit bivector is \(\pm\mathbf{j}\)\" [29].