# Vector Signals and Invariant Systems: Re-Tooling Linear Systems Theory
## 3 Notation and Terminology for Geometric Algebra
: _We represent the VV signal \(\mathbf{x}(t)\in\mathbb{L}^{n}\ \forall\ t\in\mathbb{R}\) in polar form relative to a Euclidean orthonormal basis vector \(\mathbf{e}_{1}\) as_  
\[\mathbf{B}=\mathbf{u}_{1}\wedge\mathbf{u}_{2}\wedge\cdots\wedge\mathbf{u}_{k}. \tag{18}\]5  
_where \(x(t)\) is a real \(x(t)\in\mathbb{R}\ \forall\ t\), and \(X(t)\) is a rotor formed as the sum of a scalar and bivector, i.e., \(\{X\in\mathbb{R}\oplus\mathbb{B}^{n}:\|X\|=1\}\)._  
**Remark 5**. : _For CV systems, there exists a distinguished axis (the real axis) that is used as the reference by which angles are measured. In other words, within the CV systems, the angle may be measured with respect to a unit element along the real axis in the complex plane. However, the proposed coordinate-free treatment of vector-valued systems has no such distinguished axis. Thus, we choose \(\mathbf{e}_{1}\) as our reference--essentially associating \(\mathbf{e}_{1}\) with the unit element along the real axis of the traditional CV linear systems theory._  
**Definition 14** (Totally Orthogonal Planes (TOPs)). : _Two planes \(\mathbf{j}_{1}\) and \(\mathbf{j}_{2}\) are totally orthogonal if they satisfy_  
\[\mathbf{B}=\mathbf{u}_{1}\wedge\mathbf{u}_{2}\wedge\cdots\wedge\mathbf{u}_{k}. \tag{18}\]6  
**Corollary 3** (Commutative Rotations for TOPs). : _Let \(\mathbf{j}_{1}\) and \(\mathbf{j}_{2}\) be TOPs. Then, the associated rotations represented by the simple rotors \(R_{1}=\mathrm{e}^{\mathrm{j}_{1}\theta_{1}}\) and \(R_{2}=\mathrm{e}^{\mathrm{j}_{2}\theta_{2}}\) are commutative_  
\[\mathbf{B}=\mathbf{u}_{1}\wedge\mathbf{u}_{2}\wedge\cdots\wedge\mathbf{u}_{k}. \tag{18}\]7  
**Definition 15** ((Canonical) Set of TOPs). : _We define a (canonical) set of totally orthogonal planes for \(N\)-dimensional space as_  
\[\mathbf{B}=\mathbf{u}_{1}\wedge\mathbf{u}_{2}\wedge\cdots\wedge\mathbf{u}_{k}. \tag{18}\]8_where \(M\) is given by_  
\[M = \begin{cases}\frac{N}{2},&\text{$N$ is even},\\ \frac{N+1}{2},&\text{$N$ is odd}.\end{cases}  \tag{34}\]  
_For odd \(N\)-dimensional space, analysis is performed by embedding into an even \(N+1\)-dimensional space._  
**Remark 6**. : _Embedding of an odd-dimensional space within an even-dimensional space may be considered as an extension of standard practice in the following sense. The analysis of a one-dimensional real-valued signal is conventionally performed by embedding the signal into the two-dimensional complex plane by considering it as a purely real CV signal._  
**Remark 7**. : TOPs have two important features: (1) They consist of pairwise commuting planes, which implies orthogonal (commuting) rotations [33]. This is important when considering rotations about arbitrary planes, because factoring a rotation into TOPs enables commutation of rotations in TOPs. (2) They form mutually exclusive or non-overlapping orthogonal planes in the sense that the intersection of the planes is empty. For example, in \(\mathbb{R}^{3}\) space, \(\{\mathbf{e}_{1}\mathbf{e}_{2}\), \(\mathbf{e}_{2}\mathbf{e}_{3}\), \(\mathbf{e}_{3}\mathbf{e}_{1}\}\) form a set of orthogonal planes. However, the planes \(\mathbf{e}_{1}\mathbf{e}_{2}\) and \(\mathbf{e}_{2}\mathbf{e}_{3}\) share any vector along the line of \(\mathbf{e}_{2}\) and the planes \(\mathbf{e}_{1}\mathbf{e}_{2}\) and \(\mathbf{e}_{3}\mathbf{e}_{1}\) share any vector along the line of \(\mathbf{e}_{1}\). On the other hand, embedding \(\mathbb{R}^{3}\) into \(\mathbb{R}^{4}\), we form the set of TOPs as \(\{\mathbf{e}_{1}\mathbf{e}_{2}\), \(\mathbf{e}_{3}\mathbf{e}_{4}\}\), which have no vectors in common. This is important when decomposing a signal into TOPs._  
**Definition 16** (Decomposition of a VV Signal into TOPs).