Linear Algebra

A mathematically rigorous first course in linear algebra emphasizing geometric intuition, coordinate-free thinking, and the distinction between operators and matrices.

Work in Progress

This resource is actively under development and edits occur frequently.

Content is being added chapter by chapter. Stay tuned for updates.

This resource develops linear algebra as a rigorous first course that covers standard material while setting up the language for more advanced work. The approach emphasizes:

• Geometry first, coordinates second — concepts introduced abstractly, then computed with matrices
• Operators vs matrices — clear distinction between geometric objects and their coordinate representations
  
• Dual spaces as natural — linear functionals as geometric objects, not just “row vectors”
• Building toward advanced topics — language that prepares for differential geometry, operator theory, and modern applications

Pedagogical Themes

Geometry First, Coordinates Second
Concepts are always introduced abstractly (coordinate-free), then computed using coordinates/matrices.

Operators vs Matrices
Operators are geometric objects (basis-independent); matrices are coordinate representations (basis-dependent). Similar matrices represent the same operator in different coordinates.

Building Toward Advanced Topics
The language prepares for differential geometry (tangent/cotangent bundles), operator theory, and Hilbert spaces, while staying focused on finite dimensions.

Structure and Approach

Each chapter follows a consistent progression:

1. Definitions and intuition — Concepts motivated geometrically or through linear-algebraic ideas

2. Theorems and examples — Formal statements with representative examples

3. Proofs — Collapsible proofs for deeper study

Visualizations and diagrams appear throughout, highlighting linear structures and geometric intuition.

Why This Approach?

Linear algebra is fundamental to modern mathematics and its applications, providing the language for multidimensional spaces, transformations, and structure. This course builds a solid theoretical foundation while maintaining computational skill.

Connections forward:

• Differential geometry (tangent and cotangent bundles)

• Functional analysis (infinite-dimensional vector spaces)

• Differential equations (operator methods)

• Physics (quantum mechanics, relativity)

Connections to earlier courses:
This course builds on concepts from Calculus 1-2, particularly:

• Linear approximation and differentials

• Function spaces and integration

• Sequences and convergence (foundations for infinite-dimensional spaces)

The approach is rigorous yet accessible to motivated students who have completed a solid calculus sequence.