Represent
Declare the space, basis, coordinates and measurement units.
SRAI Book 1 · Chapter 5 · PU-B01-C05
Vector Spaces, Bases, Rank and Projections
Explore span, bases, coordinates and rank, then connect orthogonal projection to least squares and reproducible computation. Learn what a representation retains, what its residual leaves out, and why geometric approximation alone is not a policy recommendation.
01 / LEARNING OUTCOMES
Understand what a representation retains and leaves out.
- Connect span, linear independence, bases and dimension.
- Express the same vector in different coordinate systems.
- Interpret column space, row space, null space and rank.
- Apply the rank-nullity relationship.
- Explain orthogonal projection and least squares geometrically.
- Verify numerical rank, projector properties and residual orthogonality.
02 / ANALYTICAL SEQUENCE
From a chosen space to a verified approximation.
Identify
Find independent directions and distinguish rank from redundant columns.
Project
Separate a vector into its approximation within the space and its perpendicular residual.
Verify
Check symmetry, idempotence and residual orthogonality with explicit tolerances.
03 / GEOMETRY AND COMPUTATION
The best approximation still leaves a residual.
Projecting (2, 3, 4) onto the first two coordinate directions gives (2, 3, 0), with residual (0, 0, 4). The squared lengths satisfy 29 = 13 + 16.
Least squares projects the response onto the design matrix's column space. The fitted response is unique, although coefficients need not be unique when the columns are dependent.
CORE CONTROLS
| Basis | Independent and spanning |
| Rank | Threshold explicitly declared |
| Orthogonal projector | Symmetric and idempotent |
| Residual | Orthogonal to the fitted space |
The notebook uses stable numerical methods and checks the results.
A projection answers a specified mathematical question.
It depends on the chosen space, scaling and distance measure.
A close fit is not automatically a recommendation.
Feasibility, constraints and domain evidence remain essential. Unconstrained projection coefficients are not automatically valid mixture weights.
04 / REPRODUCIBILITY AND RELEASE CONTROL
Run the notebook with its declared runtime.
- Notebook v0.2.2 downloads its exact runtime automatically when a local wheel is unavailable; no manual wheel upload is required.
- The runtime srai_math 1.1.1rc1 is a prerelease, pinned by SHA-256 and explicitly accepted by the owner. It has not been promoted to a stable version.
- Use a fresh Colab runtime and Run all. Internet access and the notebook's numerical dependencies are required.
- The chapter, notebook and production assets are owner-approved. This release proceeds without a separate independent review, under explicit owner authorization.
- The 16-minute video accompanies the chapter, exercises with solutions and Executive Brief.
CONTROLLED RESOURCES
Read, reproduce, practise and review.
READ
View the complete GitHub production unit ↗
Controlled Chapter 5
Controlled publication edition.
PDF ↗ REPRODUCECanonical notebook
Execution-tested and owner-approved.
IPYNB ↗ PRACTISEExercises and solutions
Progress toward independent application.
PDF ↗ RUNOpen in Google Colab
Run the approved notebook with its checksum-pinned runtime.
COLAB ↗ DOWNLOADComplete release package
Chapter, notebook, exercises, Executive Brief, runtime and validation records.
ZIP ↓ APPLYExecutive brief
Connect analytical controls to decisions.
PDF ↗