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Browse all available quizzes, quizbooks, articles, textbooks, SynLearn templates, and SynPrep practice tracks.

1890 learning items

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Quiz

MML-05-06: Matrix Condition Number, Ill-Conditioning & Numerical Regularization

30 worked single-select and multiselect questions. Tier 2–3. Prerequisites: QS 5.5. Learning goals: Compute Euclidean condition numbers from extreme singular values; Distinguish absolute scale from relative sensitivity; Apply right-hand-side perturbation bounds with fixed invertible matrix; Diagnose anisotropic noise amplification and numerical rank; Compare normal-equation conditioning with the original system; Derive ridge normal equations and singular-direction filters; Quantify regularization stability and bias tradeoffs. Column gradients and output-by-input Jacobians are used where relevant. Detailed solutions explain every choice. References: [Source 1](https://mml-book.github.io/book/mml-book.pdf), [Source 2](https://www.ee.iitm.ac.in/uday/notes/linalg/orth.html), [Source 3](https://mml-book.github.io/book/mml-book.pdf). Ridge reference: [UBC Stat406](https://ubc-stat.github.io/stat-406/schedule/lectures/lecture_06_ridge_regression.html).

mathematics

Oct 31, 2026

30 questions

Quiz

MML-05-05: Moore-Penrose Pseudoinverse: Construction, Properties & Minimum Norm

35 worked single-select and multiselect questions. Tier 2–4. Prerequisites: QS 5.4. Learning goals: Construct a pseudoinverse by reciprocating positive singular values; Check all four Moore–Penrose identities; Use full-column-rank and full-row-rank formulas with their hypotheses; Solve consistent underdetermined systems with minimum Euclidean norm; Solve inconsistent or rank-deficient least-squares systems; Interpret fitted-value and row-space projectors; Prove minimum-norm uniqueness and distinguish it from exact interpolation. Column gradients and output-by-input Jacobians are used where relevant. Detailed solutions explain every choice. References: [Source 1](https://mml-book.github.io/book/mml-book.pdf), [Source 2](https://netlib.org/lapack/lug/node27.html). Definition reference: [University of Michigan](https://socr.umich.edu/DSPA3/DSPA3_notes/DSPA3_03_LinearAlgebra_MatrixComputing_Regression.html).

mathematics

Oct 31, 2026

35 questions

Quiz

MML-05-04: Low-Rank Matrix Approximation & The Eckart-Young-Mirsky Theorem

35 worked single-select and multiselect questions. Tier 2–4. Prerequisites: QS 5.3. Learning goals: Formulate rank-constrained approximation and identify its norm; Compute spectral and Frobenius residuals from singular-value tails; Choose rank from error or energy budgets; Reconstruct low-rank matrices and compare candidate approximations; Distinguish optimality, uniqueness and repeated singular values; Derive optimality using null spaces and orthogonal dyads; Diagnose compression, denoising and semantic-retrieval limitations. Column gradients and output-by-input Jacobians are used where relevant. Detailed solutions explain every choice. References: [Source 1](https://www.math.cmu.edu/~zechengz/spring_2023_nla/svd_all.pdf), [Source 2](https://mml-book.github.io/book/mml-book.pdf).

mathematics

Oct 31, 2026

35 questions

Quiz

MML-07-03: Reverse-Mode Automatic Differentiation & Vector-Jacobian Products (VJPs)

MML-07-03: Reverse-Mode Automatic Differentiation & Vector-Jacobian Products (VJPs). 40 conceptual, computational and ML diagnostic questions. Prerequisites: QS 7.2. Intended cognitive levels: Tier 2–4. Includes single-select and multiselect MCQs with worked explanations. All derivatives use real variables; scalar vector-gradients are columns and matrix gradients use the Frobenius inner product unless explicitly stated.

mathematics

Oct 31, 2026

40 questions

Quiz

MML-07-02: Forward-Mode Automatic Differentiation & Dual Numbers

MML-07-02: Forward-Mode Automatic Differentiation & Dual Numbers. 30 conceptual, computational and ML diagnostic questions. Prerequisites: QS 7.1. Intended cognitive levels: Tier 2–4. Includes single-select and multiselect MCQs with worked explanations. All derivatives use real variables; scalar vector-gradients are columns and matrix gradients use the Frobenius inner product unless explicitly stated.

mathematics

Oct 31, 2026

30 questions

Quiz

MML-07-01: Computational Graphs: Directed Acyclic Graphs (DAGs) & Node Operations

MML-07-01: Computational Graphs: Directed Acyclic Graphs (DAGs) & Node Operations. 25 conceptual, computational and ML diagnostic questions. Prerequisites: Part 06. Intended cognitive levels: Tier 1–2. Includes single-select and multiselect MCQs with worked explanations. All derivatives use real variables; scalar vector-gradients are columns and matrix gradients use the Frobenius inner product unless explicitly stated.

mathematics

Oct 31, 2026

25 questions

Quiz

MML-06-09: Multivariate Taylor Expansions & Local Quadratic Approximations

MML-06-09: Multivariate Taylor Expansions & Local Quadratic Approximations. 30 conceptual, computational and ML diagnostic questions. Prerequisites: QS 6.8. Intended cognitive levels: Tier 2–3. Includes single-select and multiselect MCQs with worked explanations. All derivatives use real variables; scalar vector-gradients are columns and matrix gradients use the Frobenius inner product unless explicitly stated.

mathematics

Oct 31, 2026

30 questions

Quiz

MML-06-08: Second Partial Derivatives, The Hessian Matrix & Schwarz's Theorem

MML-06-08: Second Partial Derivatives, The Hessian Matrix & Schwarz's Theorem. 35 conceptual, computational and ML diagnostic questions. Prerequisites: QS 6.7. Intended cognitive levels: Tier 2–4. Includes single-select and multiselect MCQs with worked explanations. All derivatives use real variables; scalar vector-gradients are columns and matrix gradients use the Frobenius inner product unless explicitly stated.

mathematics

Oct 31, 2026

35 questions

Quiz

MML-06-07: Matrix Calculus: Gradients of Inverses, Determinants & Log-Determinants

MML-06-07: Matrix Calculus: Gradients of Inverses, Determinants & Log-Determinants. 30 conceptual, computational and ML diagnostic questions. Prerequisites: QS 6.6. Intended cognitive levels: Tier 3–4. Includes single-select and multiselect MCQs with worked explanations. All derivatives use real variables; scalar vector-gradients are columns and matrix gradients use the Frobenius inner product unless explicitly stated.

mathematics

Oct 31, 2026

30 questions

Quiz

MML-06-06: Matrix Calculus: Gradients of Traces, Linear Forms & Quadratic Forms

MML-06-06: Matrix Calculus: Gradients of Traces, Linear Forms & Quadratic Forms. 40 conceptual, computational and ML diagnostic questions. Prerequisites: QS 6.5, Part 04. Intended cognitive levels: Tier 2–4. Includes single-select and multiselect MCQs with worked explanations. All derivatives use real variables; scalar vector-gradients are columns and matrix gradients use the Frobenius inner product unless explicitly stated.

mathematics

Oct 31, 2026

40 questions

Quiz

MML-06-05: Multivariable Chain Rule for Scalar, Vector & Matrix Expressions

MML-06-05: Multivariable Chain Rule for Scalar, Vector & Matrix Expressions. 40 conceptual, computational and ML diagnostic questions. Prerequisites: QS 6.4. Intended cognitive levels: Tier 2–4. Includes single-select and multiselect MCQs with worked explanations. All derivatives use real variables; scalar vector-gradients are columns and matrix gradients use the Frobenius inner product unless explicitly stated.

mathematics

Oct 31, 2026

40 questions

Quiz

MML-12-04: Gradient Descent: Step Size, Lipschitz Smoothness & Convergence Proofs

Study Gradient Descent: Step Size, Lipschitz Smoothness & Convergence Proofs. Master rigorous derivations, step-by-step numerical computations, multiselect conceptual checks, and diagnostic applications in machine learning models. Prerequisites and cognitive tiers follow the master syllabus.

mathematics

Oct 31, 2026

40 questions