Represent
Encode an observation, state or direction as ordered coordinates with declared meaning.
SRAI Book 1 · Chapter 3 · PU-B01-C03
Vector Foundations
Vectors transform ordered information into geometry. Lesson 3 develops vector representation, arithmetic, norms, distance, dot products, cosine similarity, projection, orthogonality and scaling, then connects these foundations to statistics, data science, artificial intelligence and transparent decision systems.
01 / LEARNING OUTCOMES
Turn ordered information into inspectable geometry.
- Distinguish scalars, points and vectors.
- Compute vector arithmetic and common norms.
- Interpret distance, dot products, angles and cosine similarity.
- Compute projections and verify orthogonal rejections.
- Explain span, bases and Gram-Schmidt orthogonalization.
- Document coordinate order, units, scaling, weighting and missing-data rules.
02 / GEOMETRIC STRUCTURE
Four operations connect representation to analytical meaning.
Measure
Use norms and distance to quantify magnitude and separation under a chosen geometry.
Compare
Use dot products, angles and cosine similarity to examine directional alignment.
Decompose
Project onto chosen directions and verify the orthogonal component independently.
03 / VECTOR OPERATIONS
Simple operations support high-dimensional reasoning.
Vector addition and scalar multiplication combine and rescale states. Norms formalize different meanings of size, while the dot product connects algebra to angle and alignment.
Projection separates the component aligned with a declared direction from the orthogonal rejection. The notebook verifies this decomposition through the invariant uTr = 0.
CORE CONTROLS
| Dimension | Coordinate order must agree |
| Normalization | Zero vectors are invalid |
| Projection | Direction must be nonzero |
| Orthogonality | Residual dot product is zero |
Every calculation is interpreted under an explicit representation.
A score describes the chosen representation.
Distance and similarity depend on coordinate definitions, scale, direction, weights and the metric selected.
Meaning requires evidence beyond the score.
High cosine similarity does not by itself establish semantic identity, equal welfare, causation or interchangeable policy needs.
04 / DEFENSIBLE VECTOR SYSTEMS
Geometry becomes evidence only after its construction is justified.
- Declare coordinate names, order, units and reference period.
- Record scaling, normalization, indicator direction and weights.
- Keep missing, uncertain and not-applicable values visible.
- Test zero vectors, incompatible dimensions and non-finite values.
- Compare conclusions under plausible alternative transformations.
- Preserve the representation version and limits of interpretation.
CONTROLLED RESOURCES
Read, reproduce, practise and review.
READ
View the complete GitHub production unit ↗
Audited Chapter 3
Controlled publication edition.
PDF ↗ REPRODUCECanonical notebook
Executable and independently verified.
IPYNB ↗ PRACTISEExercises and solutions
Progress toward independent application.
PDF ↗ RUNOpen in Google Colab
Execute the approved notebook in a clean cloud runtime.
COLAB ↗ DOWNLOADComplete release package
Chapter, notebook, exercises, Executive Brief, code, metadata and validation records.
ZIP ↓ APPLYExecutive brief
Connect analytical controls to decisions.
PDF ↗