MA1034: Process Modelling using Linear Algebra

Session 07: Orthogonality

Prof. Alejandro Ucan-Puc

Tecnológico de Monterrey • Departamento de Matemáticas

Goals of the Session

  • Introduce the concept of Inner Product and vector norm.
  • Calculate angles and distances between vectors in engineering models.
  • Define vector orthogonality ($\langle u, v \rangle = 0$).
  • Compute orthogonal projections onto subspaces.

Motivation: Product Quality Control

A factory evaluates batch quality metrics $q = (2.3, 1.7, 0.8)^T$ along three independent quality directions:

  • $b_1 = (1, 1, 1)^T$ (Balanced quality across all metrics)
  • $b_2 = (1, -1, 0)^T$ (Precision vs. Durability trade-off)
  • $b_3 = (1, 1, -2)^T$ (Finish emphasis)

How do we project batch quality onto these orthogonal directions?

Inner Product & Norm

Definition: For $u, v \in \mathbb{R}^n$, the standard inner product (dot product) is:

$$\langle u, v \rangle = u^T v = \sum_{i=1}^n u_i v_i$$

Norm (Magnitude): $\|u\| = \sqrt{\langle u, u \rangle} = \sqrt{\sum_{i=1}^n u_i^2}$.

Angle $\theta$: $\cos \theta = \frac{\langle u, v \rangle}{\|u\| \|v\|}$.

Orthogonality

Definition: Two vectors $u, v \in V$ are orthogonal if their inner product is zero:

$$\langle u, v \rangle = 0$$

Pythagorean Theorem: $\|u + v\|^2 = \|u\|^2 + \|v\|^2 \iff \langle u, v \rangle = 0$.

Orthogonal Projection

Definition: The orthogonal projection of a vector $u$ onto a non-zero vector $v$ is:

$$\text{proj}_v(u) = \frac{\langle u, v \rangle}{\langle v, v \rangle} \, v$$

The residual $u - \text{proj}_v(u)$ is orthogonal to $v$.

Summary & Next Step

Orthogonality minimizes redundant information and simplifies projections in data analysis.

Next Session: The Gram-Schmidt Process and Orthonormal Bases.