Open workspace
Track 3 of 5 · Beginner

Mathsfor reading and building models.

Each idea is drawn first, then computed by hand, then used on a model.

Start with 4.1 DifferentiationSee the syllabus
5modules
21lessons planned
1ready to study now
1free video

The syllabus5 modules, one objective per lesson.

The course is being built in this order. Every lesson is listed, including the ones still in production, so you can see where the track goes. Open lessons have their parts, key terms and quiz free.

Module 1

Reading Maths

3 lessons, in production

  1. 1.1Functions and Their GraphsRead a function as a machine from input to output, and recognise the shapes ML keeps using.
    What it will cover
    • Input, output, and the graph as a picture of the rule
    • Linear, quadratic, exponential and logarithmic shapes
    • Slope as a rate of change (the idea video 2 makes exact)
    • Shifting and stretching a graph
    In production
  2. 1.2Exponents and LogarithmsKnow why logs appear everywhere in ML: they turn products into sums, and huge ranges into small ones.
    What it will cover
    • Powers and roots; e
    • The log as the inverse of the exponential
    • log(ab) = log a + log b, and why that matters for likelihoods
    • Log scales on plots; the log transform of a skewed feature
    In production
  3. 1.3Reading ML NotationRead the formulas in ML papers and docs without stalling.
    What it will cover
    • Subscripts and superscripts: xᵢ, x⁽ʲ⁾, wⱼ
    • Σ and Π
    • Hats, bars and stars: ŷ, x̄, w*
    • Vectors bold, matrices capital; argmin and argmax
    • Set notation: ∈, ℝᵈ
    In production
Module 2

Vectors

5 lessons, in production

  1. 2.1VectorsSee a row of data as a point and as an arrow, in any number of dimensions.
    What it will cover
    • 2-D, 3-D and d-D vectors
    • Row and column vectors
    • A dataset as a cloud of points
    In production
  2. 2.2Vector Arithmetic and NormsAdd, scale and measure vectors, and know that "length" has more than one definition.
    What it will cover
    • Addition, scalar multiplication, the mean of vectors
    • L2 norm, L1 norm, L∞ norm
    • Distance as the norm of a difference
    • The norms as penalties (the link to ridge and lasso)
    In production
  3. 2.3The Dot Product, Angle and ProjectionCompute a dot product and read it as alignment, and project one vector onto another.
    What it will cover
    • The dot product, algebraic and geometric
    • The angle between vectors; cosine similarity
    • Unit vectors
    • Projection onto a direction
    In production
  4. 2.4Lines, Planes and HyperplanesRead `wᵀx + b = 0` as a boundary in any dimension: which side a point is on, and how far away it is.
    What it will cover
    • The line in 2-D; the plane in 3-D; the hyperplane in d-D
    • w as the normal vector; b as the offset
    • Signed distance from a point to a hyperplane
    • Half-spaces: the sign as the class
    In production
  5. 2.5Circles, Spheres, Ellipses and Boxes in d DimensionsWrite the shapes that show up as decision regions and as constraints, in any dimension.
    What it will cover
    • Circle, sphere, hypersphere
    • Ellipse, ellipsoid, and the axis lengths
    • Square, cube, hypercube; axis-parallel boxes (tree regions)
    • The corners of a hypercube and the curse of dimensionality
    In production
Module 3

Matrices

7 lessons, in production

  1. 3.1Matrices and the Matrix–Vector ProductSee a dataset as a matrix, and a linear model's predictions as one matrix–vector product.
    What it will cover
    • Shape (n × d); rows as samples, columns as features
    • Xw: every prediction at once, as many dot products
    • The column mean and the centred matrix
    In production
  2. 3.2Matrix Multiplication as TransformationRead a matrix as something that moves space: rotate, stretch, squash.
    What it will cover
    • Multiplying matrices, and the shape rule
    • A 2 × 2 matrix acting on the plane, animated
    • Composition as multiplication; why order matters
    In production
  3. 3.3Transpose, Identity and InverseUse the transpose and inverse, and know when an inverse does not exist.
    What it will cover
    • The transpose, and (AB)ᵀ = BᵀAᵀ
    • The identity matrix
    • The inverse as undoing a transformation
    • Singular matrices: the squashed dimension
    In production
  4. 3.4Linear Systems, Rank and Least SquaresSolve `Ax = b` exactly when it can be solved, and in the least-squares sense when it cannot. Derive the normal equations video 5 used.
    What it will cover
    • Systems of equations as matrix equations
    • Rank, and dependent columns (collinearity, seen again)
    • Least squares: projecting y onto the column space
    • XᵀXw = Xᵀy, derived; solve rather than invert
    In production
  5. 3.5Eigenvalues and EigenvectorsFind the directions a matrix only stretches, and read the eigenvalues as how much.
    What it will cover
    • Av = λv, animated as vectors that keep their direction
    • Symmetric matrices: real eigenvalues, perpendicular eigenvectors
    • Eigen-decomposition
    • Curvature directions (the link to ML 3.5)
    In production
  6. 3.6Singular Value DecompositionFactor any matrix into rotate–stretch–rotate, and keep only the largest parts.
    What it will cover
    • A = UΣVᵀ, in pictures
    • Singular values as importance
    • Low-rank approximation
    • SVD as the engine behind PCA and recommenders
    In production
  7. 3.7The Covariance MatrixBuild the covariance matrix from a centred data matrix and read its entries and its shape.
    What it will cover
    • (1/n) XᵀX for centred X
    • Diagonal entries as variances; off-diagonal entries as covariances
    • Its ellipse: the shape of the data cloud
    • Standardising first: the correlation matrix
    In production
Module 4

Calculus

1 of 5 lessons open

  1. 4.1DifferentiationFree video
  2. 4.2Convexity, Local Minima and Saddle PointsTell a bowl from a landscape, and know what a zero gradient does and does not guarantee.
    What it will cover
    • Convex functions and convex sets
    • Why convex losses (squared, log, hinge) have one minimum
    • Local minima and saddle points in non-convex functions
    • Why initialisation starts to matter (K-means, EM, later neural networks)
    In production
  3. 4.3Curvature: Second Derivatives, Taylor and NewtonUse the second derivative to see how a function bends, and step straight to the bottom of a local quadratic.
    What it will cover
    • The second derivative and curvature
    • The Hessian: curvature in every direction
    • Taylor approximation, first and second order
    • Newton's method against gradient descent
    In production
  4. 4.4Constrained Optimisation and Lagrange MultipliersMinimise a function subject to a constraint, and read the multiplier.
    What it will cover
    • Constraints as curves and surfaces; the tangency picture
    • The Lagrangian
    • Inequality constraints and KKT, named
    • The constraint form of ridge and lasso: the ball and the diamond
    In production
  5. 4.5Integrals as AreaRead an integral as accumulated area, enough for densities, CDFs and AUC.
    What it will cover
    • Area under a curve as a sum of thin strips
    • The integral as the inverse of the derivative, pictured
    • Probability as area under a density
    • AUC as an area
    In production
Module 5

Numbers on a Computer

1 lessons, in production

  1. 5.1Floating Point and Numerical StabilityKnow why a correct formula can give a wrong answer on a computer, and the standard fixes.
    What it will cover
    • Floating-point numbers, rounding, 0.1 + 0.2
    • Overflow and underflow
    • Products of small probabilities → sums of logs
    • Log-sum-exp; the numerically stable sigmoid and softmax
    In production

What you’ll learnby the end of the track.

  • Read machine learning notation aloud in words.
  • Use vectors, dot products and matrices to describe data.
  • Find a derivative and a gradient, and say what it means.
  • Follow the maths behind a model's training loop.

Before you startand how the lessons work.

  • School algebra. Each lesson rebuilds what it needs before using it.
  • Every open lesson shows its parts, key terms and quiz for free. Sign in to check your answers and save your progress.
  • While the course is being recorded, the practice problems, common mistakes, self-checking notebooks, cheat sheets and project checks are free with an account too.

Where it leadslive roles this track prepares you for.

Openings from companies’ own career pages, updated continuously.

Questionsabout this track.

How much maths do I need for machine learning?

Linear algebra, a little calculus and some probability. This track covers the parts models actually use, just before the lessons that need them.

Do I need to finish this track before machine learning?

No. Each maths lesson is best taken just before the machine learning lesson that uses it. The learning path says which.

The other tracksand the order to take them in.

See the learning path