Maths tutorials and worked examples

Trace matrices, dimensions and linear maps.

• Matrix multiplication with shapes: map a batch of vectors

  Trace a 2×3 batch through a 3×2 matrix, calculate individual entries and distinguish matrix multiplication from broadcasting and elementwise products.

  tutorial · 4 min read

Derive model gradients and check them numerically.

• Derivatives for machine learning: check a squared-error gradient

  Derive the weight and bias gradients of a linear model, verify dimensions and compare central differences with the analytic result.

  tutorial · 4 min read

Connect update rules, curvature and convergence.

• Gradient descent learning rates: convergence, oscillation and divergence

  Run three learning rates on an explicit quadratic, derive the stability boundary and compare reproducible objective curves.

  tutorial · 4 min read

Use geometric interpretations to choose and check similarity measures.

• Cosine similarity vs Euclidean distance: choose what closeness means

  Calculate both measures on the same vectors, explain their different rankings, and connect normalized distance to cosine similarity.

  comparison · 4 min read

Evaluate mathematical expressions reliably in finite precision.

• Log-sum-exp: calculate model scores without exponential overflow

  Derive the maximum-shift identity, reproduce overflow and verify stable log-sum-exp and softmax calculations in Python.

  tutorial · 4 min read

Connect prerequisites to the model calculations they explain.

• Prerequisites for machine learning and AI: the maths, statistics and Python you need

  Machine learning and AI prerequisites: the Python, maths and statistics you need, what deep learning and LLMs add, and a checklist to test whether you are ready.

  learning path · 9 min read