6  Inner Product Spaces

In this chapter, we explore the concept of inner product spaces and their fundamental properties. A mind map provides a structured overview of the key topics discussed:

Figure 6.1: Mind Map of Inner Product Spaces

Inner product spaces extend vector spaces by introducing a notion of length and angle between vectors. They provide a structured way to measure similarity, correlation, and projections of data represented as vectors.

In the context of mining engineering:

• Ore Similarity: Ore grades at different locations can be represented as vectors in \(\mathbb{R}^n\). The inner product allows computation of similarity measures between samples, aiding in ore classification and blending chiles2012?, rubinstein2016?.
  
• Sensor Signal Analysis: Signals from monitoring equipment or geotechnical sensors can be treated as vectors. Inner products enable correlation analysis, anomaly detection, and pattern recognition [1].
  
• Resource Allocation Optimization: Tasks such as distributing equipment, workforce, and materials can be modeled as vectors, and inner products can support least-squares optimization to minimize cost or maximize efficiency meyer2000?.
  
• Spatial Estimation / Geostatistics: Inner products underlie kriging and other spatial interpolation methods, enabling accurate estimation of mineral concentrations at unsampled locations chiles2012?.
  
• 3D Modeling and Projections: Positions of boreholes, tunnels, or ore bodies in \(\mathbb{R}^3\) can be analyzed using inner products to compute angles, distances, and orthogonal projections, useful for mine planning and visualization rubinstein2016?.

Inner product spaces provide a rigorous way to quantify length, similarity, and orthogonality of vectors, which is crucial for optimization, geostatistical modeling, and operational decision-making in mining. They extend the utility of vector spaces by enabling precise measurement and projection operations that underpin advanced analytics and planning [1], meyer2000?, chiles2012?, rubinstein2016?.

6.1 Definition

An inner product space is a vector space \(V\) equipped with an inner product \(\langle \cdot, \cdot \rangle : V \times V \to \mathbb{R}\) (or \(\mathbb{C}\)) that satisfies:

1. Positivity:
   \[ \langle v, v \rangle \ge 0, \quad \forall v \in V \]
   and \(\langle v, v \rangle = 0 \iff v = 0\).

2. Linearity in the first argument:
   \[ \langle a u + b v, w \rangle = a \langle u, w \rangle + b \langle v, w \rangle \]
   for scalars \(a, b\) and vectors \(u, v, w \in V\).

3. Conjugate symmetry (for complex spaces):
   \[ \langle u, v \rangle = \overline{\langle v, u \rangle} \]

4. Real symmetry (for real spaces):
   \[ \langle u, v \rangle = \langle v, u \rangle \]

6.2 Properties

The inner product induces several important properties:

• Norm / Length:
  \[ \|v\| = \sqrt{\langle v, v \rangle} \]

• Distance:
  \[ d(u, v) = \|u - v\| \]

• Angle between vectors:
  \[ \cos \theta = \frac{\langle u, v \rangle}{\|u\| \|v\|} \]

• Orthogonality:
  Two vectors \(u, v\) are orthogonal if \(\langle u, v \rangle = 0\).

• Projection:
  The projection of \(u\) onto \(v\) is
  \[ \text{proj}_v(u) = \frac{\langle u, v \rangle}{\langle v, v \rangle} v \]

6.3 Examples

1. Euclidean space \(\mathbb{R}^n\)
   Standard inner product:
   \[ \langle u, v \rangle = \sum_{i=1}^{n} u_i v_i \]

2. Function space \(L^2[a,b]\)
   \[ \langle f, g \rangle = \int_a^b f(x) g(x) dx \]

3. Polynomial space
   \[ \langle p, q \rangle = \sum_{i=0}^n p(i) q(i) \]

4. Matrix space
   \[ \langle A, B \rangle = \text{trace}(A^T B) \]

6.4 Applications

• Ore Similarity and Blending:
  Vectors represent ore grades at different locations. Inner products measure similarity for classification and blending strategies.

• Sensor Data Analysis:
  Signals from monitoring sensors can be represented as vectors. Inner products allow correlation and anomaly detection.

• Resource Allocation Optimization:
  Allocation of equipment, labor, and materials can be modeled as vectors. Inner products are used in least-squares optimization to minimize costs or maximize efficiency.

• Spatial Estimation / Geostatistics:
  Inner products underpin kriging and other interpolation methods for estimating mineral concentrations at unsampled locations.

• 3D Modeling and Projections:
  Positions of boreholes, tunnels, and ore bodies in \(\mathbb{R}^3\) allow calculations of angles, distances, and orthogonal projections, aiding mine planning and visualization.

• Risk Assessment:
  Safety indices across mine zones can be represented as vectors. Norms and projections help quantify risk and guide mitigation strategies.

References

[1]

Golub, G. H. and Loan, C. F. V., Matrix computations, Johns Hopkins University Press, Baltimore, MD, 2013