Compute vector and matrix operations used in machine learning models.

Interpret dot products, norms, cosine similarity, and vector projections geometrically.

Perform eigendecomposition of a small symmetric matrix and interpret its eigenvectors and eigenvalues as transformation directions and scales.

Apply singular value decomposition to a small matrix and interpret its low-rank approximation.

Compute partial derivatives and gradients of multivariable functions.

Apply the chain rule through a simple computational graph and use the resulting gradient to perform a gradient-descent update.

Use conditional probability and Bayes’ rule to update probabilities from evidence.

Calculate expectation, variance, and covariance for random variables.