🧮 Mathematics for Machine Learning 1: Linear Algebra Foundations

By Better Wise · Generated by Everwise

Build the linear algebra intuition behind machine learning: see vectors and matrices as transformations, then compute eigenvalues, projections and SVD yourself in NumPy.

Level
Beginner
Goal
Fundamentals
Language
English
Focus
vectors and norms matrix operations and transformations eigenvalues and SVD geometric intuition NumPy practice
Prior knowledge
High-school algebra (equations, functions), Basic Python or willingness to run short NumPy snippets
Practice
Guided exercises
Time
~4.8h
Voice
a patient university lecturer who draws every operation as a geometric transformation before touching the algebra
Not covered
Calculus and derivatives, Probability and statistics, Deep learning models

What you will achieve

By the end of this course you will be able to look at a vector or a matrix and see a picture, not just a grid of numbers. You will build the geometric intuition first, then confirm it with short NumPy snippets you run yourself.

Concretely, you will be able to:

  • Describe a vector as an arrow, as a point, and as a NumPy array, and move between those three views without confusion.
  • Measure the size of a vector with norms and the distance between two vectors, and explain why different norms give different answers.
  • Compute a dot product and a cosine similarity, and use them to judge how alike two vectors are, the core operation behind many recommendation and search systems.
  • Read a matrix as a transformation of space, and read matrix multiplication as running one transformation after another.
  • Use the inverse, rank, and span to reason about whether a system of equations has a solution, and how many.
  • Change basis: describe the same vector in a different set of axes and understand what stays the same.
  • Project one vector onto another and see the least squares idea as the best approximation when no exact answer exists.
  • Find eigenvectors, understand eigen-decomposition, and take a first look at SVD as a way to break a matrix into simple, interpretable pieces.

Every one of these abilities is practiced in NumPy, so the ideas stick to real computation.

Who this course is for

This is a beginner course. It assumes two things only:

  • High-school algebra: you are comfortable with equations and functions.
  • Basic Python, or the willingness to run short NumPy snippets. You do not need prior NumPy experience; the snippets are small and explained as we go.

You do not need a linear algebra background, a statistics background, or any calculus. If you have met matrices before but never felt you understood them, this course is designed for exactly that gap.

Course roadmap

The course runs across three modules and twelve lessons.

Module 1: Vectors, Norms and Similarity

  • 1.1 Vectors as Arrows, Points, and NumPy Arrays. The same object seen three ways, so the geometry and the code line up.
  • 1.2 Norms and Distances: Measuring Vector Size. How to put a number on length and closeness, and when each norm is the right tool.
  • 1.3 Dot Product and Cosine Similarity. The dot product as a measure of alignment, and cosine similarity as its normalized form.
  • 1.4 Checkpoint: Vectors and Similarity. A short quiz to confirm the first module before moving on.

Module 2: Matrices and Transformations

  • 2.1 Matrices as Transformations and Matrix Multiplication as Composition. A matrix as an action on space, and multiplication as chaining actions.
  • 2.2 Inverse, Rank, and Span. What it means to undo a transformation, and how rank and span describe what a matrix can and cannot reach.
  • 2.3 Basis and Change of Basis. Choosing a different set of axes and translating cleanly between them.
  • 2.4 Checkpoint: Matrices and Transformations. A quiz covering the second module.

Module 3: Projections, Eigenvectors and SVD

  • 3.1 Projections and Least Squares Intuition. Dropping a vector onto a line or plane, and why the closest fit is often the best answer.
  • 3.2 Eigenvectors and Eigen-Decomposition. The special directions a transformation only stretches, and how to find them.
  • 3.3 A First Look at SVD. A first pass at the singular value decomposition, the workhorse behind compression and dimensionality reduction.
  • 3.4 Checkpoint: Projections, Eigen and SVD. A final quiz to close the course.

How the course works

Each lesson follows a steady rhythm. The lecturer draws the operation as a geometric transformation before any algebra appears, then you see the same idea in a short NumPy snippet, then you practice it in a guided exercise. Theory is kept short and always paired with something you run or draw.

The course mixes four formats:

  • Theory lessons that build the geometric picture with short explanations and visuals.
  • Practice exercises where you apply each concept in NumPy, guided step by step.
  • Checkpoint quizzes at the end of each module to test recall and catch gaps early.
  • A final checkpoint that ties projections, eigenvectors, and SVD together.

The total pacing estimate is about 290 minutes across 12 lessons, roughly 20 to 30 minutes per lesson. You can finish the course in a few focused sittings or spread it over a couple of weeks.

How to use it: the modules build on each other, so the intended path is linear. Module 2 assumes the vector and norm ideas from Module 1, and Module 3 assumes matrix transformations from Module 2. If you already know a topic, you can skim its theory and go straight to the exercise and checkpoint quiz to confirm you are solid. If a checkpoint quiz surprises you, revisit that lesson before continuing, because the next module will lean on it.

Start with lesson 1.1, keep a Python notebook open, and run every snippet as you meet it. The geometry will do the rest.