Linear Algebra in Any Dimension: Linear Systems and Matrices (with Python)

Linear algebra is often introduced through small, concrete examples—then quickly becomes difficult to scale when dimensions grow. This course, Linear Algebra in Any Dimension – Linear Systems and Matrices, is designed to keep the core ideas clear while moving naturally toward higher-dimensional settings and practical computation.

Why this topic matters

Many problems in science, engineering, economics, and data analysis reduce to a simple-looking statement:

Ax=b

Behind this compact form lies a powerful framework:

• a model (the system of linear equations),
• a structure (the matrix A),
• and a set of methods to determine when solutions exist, how many there are, and how to compute them efficiently.

Course focus

This course emphasizes both conceptual understanding and algorithmic practice. The progression is built around three pillars:

1) From equations to matrix form Students learn to translate a system of linear equations into:

• vector form and matrix notation,
• augmented matrices,
• and elementary row operations (with interpretation, not just mechanics).

2) Solving linear systems effectively We focus on standard, robust methods used across applied mathematics:

• Triangular systems Solving UX=Y or LY=B via backward/forward substitution.
• Gaussian elimination Elimination to obtain a triangular (or echelon) form, then back-substitution. Practical attention is given to pivot issues (e.g., zero pivots and row permutations).
• LU factorization Decomposing (A) into (L) (lower triangular) and (U) (upper triangular), so that:

3) Python as a learning tool Rather than treating computation as a “black box,” the course uses Python scripts to:

• reproduce each algorithm step by step,
• validate results through simple checks (e.g., residual (|Ax-b|)),
• and connect mathematical reasoning to implementation choices.

Who this course is for

• Undergraduate students in science, engineering, or computer science.
• Learners who want a structured approach that goes beyond “plug-and-chug” elimination.
• Instructors looking for material that blends theory, exercises, and computational activities.

What students should be able to do by the end

• Write and interpret systems in matrix form (Ax=b).
• Perform and explain elementary row operations.
• Solve triangular systems reliably (forward/backward substitution).
• Apply Gaussian elimination and interpret outcomes.
• Use LU decomposition to solve systems efficiently, especially in repeated-solve contexts.

How it fits into a curriculum

The content can support lectures, tutorials, or lab sessions:

• as a standard linear algebra unit on systems and matrices,
• or as part of a “math + scientific computing” module.

Materials typically include structured notes, guided exercises, and Python scripts (Gaussian elimination, LU, and variants), allowing instructors to adapt the course to different time constraints and student profile.

Mathedu offers institutions a lifetime license – one single purchase – that lets you enroll as many students as you want, with no caps and no recurring fees.