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Quiz

7.6-09 PageRank as a model

PageRank as a model Build a small PageRank model from link probabilities, a dangling-page rule and a teleportation distribution. Interpret the resulting stationary probabilities within the stated random-surfer model. What you will practise Create normalized outgoing-link columns Construct a damped transition matrix Handle pages without outgoing links Solve and interpret a stationary ranking Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review column-stochastic matrices and stationary distributions. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The final two-page calculation combines damping and balance equations, while interpretation questions separate visitation from claims about information quality. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

linear algebra

Oct 20, 2026

8 questions

Quiz

7.6-08 Markov convergence and Perron–Frobenius

Markov convergence and Perron–Frobenius Understand the conditions behind finite Markov-chain convergence. Use regularity, irreducibility and aperiodicity carefully, and examine counterexamples where a stationary distribution exists without convergence from every start. What you will practise Recognize positive matrix powers Distinguish uniqueness from attraction Diagnose periodic behavior Use self-loops and spectral information Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review stationary distributions, matrix powers and eigenvalue modulus. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The final computation verifies regularity directly through a positive matrix square rather than through an unsupported shortcut. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

linear algebra

Oct 20, 2026

10 questions

Quiz

7.6-07 Stationary distributions

Stationary distributions Solve for stationary distributions by combining the fixed-vector equation with probability constraints. Compare unique, nonunique and periodic examples and derive a general two-state balance formula. What you will practise Solve stationary equations Normalize by total mass Check nonnegativity and uniqueness Separate stationary states from convergence Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review column-stochastic matrices, null spaces and linear systems. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The later problems examine convex mixtures, repeated transition columns and a parameterized two-state chain. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

linear algebra

Oct 20, 2026

10 questions

Quiz

7.6-06 Column-stochastic matrices

Column-stochastic matrices Construct and use transition matrices with one consistent column-vector convention. Interpret individual entries, preserve probability mass, compose updates in the correct order and recognize absorbing states. What you will practise Build destination columns Check stochastic constraints Compute one-step distributions Compose transition matrices Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review probability vectors and matrix multiplication. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The final example tracks remaining probability at a transient state next to an absorbing state. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

linear algebra

Oct 20, 2026

8 questions

Quiz

7.6-05 Phase portraits

Phase portraits Classify planar discrete phase portraits from their eigenstructure. Identify sinks, sources, saddles and spirals, then compare the discrete interpretation with the continuous model using the same matrix. What you will practise Classify contracting and expanding modes Find stable directions of saddles Interpret turning and sign reversal State which time convention is used Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review discrete eigenmode evolution and complex conjugate pairs. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The final question determines the exact set of initial states approaching a saddle rather than classifying every trajectory identically. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

linear algebra

Oct 20, 2026

8 questions

Quiz

7.6-04 Discrete growth and decay

Discrete growth and decay Predict long-run discrete behavior from eigenvalue moduli under explicit diagonalizability assumptions. Distinguish decay, growth, bounded oscillation and convergence, while accounting for the initial state. What you will practise Use modulus rather than sign alone Identify stable eigenspaces Separate boundedness from convergence Check complex eigenvalue magnitudes Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review matrix powers, eigenmodes and complex modulus. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The last case challenges reliance on the real part of a complex eigenvalue when the model advances in discrete steps. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

linear algebra

Oct 20, 2026

8 questions

Quiz

7.6-03 Expanding an initial state

Expanding an initial state Resolve an initial state into eigen-coordinates and evolve each component independently. Connect exact coordinate solving with decay, fixed modes, alternating behavior and absent growing modes. What you will practise Solve for initial mode coefficients Evolve and recombine the modes Find limits and subsequential limits Account for zero initial coefficients Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review eigenbases, small linear systems and powers. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The final computation uses a nonorthogonal eigenbasis, making the initial coordinate conversion essential. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

linear algebra

Oct 20, 2026

8 questions

Quiz

7.6-02 Fibonacci as a matrix recurrence

Fibonacci as a matrix recurrence Model the Fibonacci recurrence as a two-component matrix update and recover its closed form from eigenmodes. The progression emphasizes state ordering, initial data and the distinction between a general recurrence and one specific sequence. What you will practise Construct the recurrence matrix Find its characteristic roots Match initial data to mode coefficients Interpret the dominant growth contribution Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review recurrences, diagonalization and quadratic roots. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. A final change of initial data shows how the same update matrix can produce a different closed-form sequence. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

linear algebra

Oct 20, 2026

8 questions

Quiz

7.6-01 Fast powers through diagonalization

Fast powers through diagonalization Use an eigenbasis to compute large matrix powers efficiently and understand why the formula works. Combine diagonal powers with basis changes, then apply the method to concrete two-by-two matrices and special spectra. What you will practise Cancel intermediate basis conversions Raise diagonal entries to powers Handle zero and negative exponents correctly Reconstruct an explicit power formula Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review diagonalization and matrix inverses. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The final structural items use powers of eigenvalues to identify matrix identities without repeated multiplication. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

linear algebra

Oct 20, 2026

8 questions

Quiz

7.B-04 Exponential series

Exponential series Introduce the exponential power series and its matrix counterpart through finite approximations. Focus on factorial coefficients, the identity term, matrix powers and the special cases where a truncation becomes exact. What you will practise Read scalar exponential terms Form matrix partial sums Recognize nilpotent termination Describe approximation and convergence Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review matrix multiplication, factorial notation and scalar exponentials. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The final approximation question checks the value, derivative and first omitted term near time zero. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

linear algebra

Oct 20, 2026

6 questions

Quiz

7.B-03 Exponential differential equations

Exponential differential equations Prepare for linear differential systems through scalar exponential solutions. Verify derivatives and initial conditions, interpret growth and decay, and recognize an eigenvector multiplied by an exponential as a vector-valued solution. What you will practise Differentiate exponential solutions Enforce an initial condition Interpret the sign of a real rate Verify a continuous eigenmode Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review elementary differentiation and the scalar exponential function. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The later items distinguish the initial coefficient from the exponential rate and check both pieces of an initial-value problem. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

linear algebra

Oct 20, 2026

6 questions

Quiz

7.B-02 Recurrences and initial states

Recurrences and initial states Translate scalar recurrences into evolving states with clearly specified initial data. The exercises connect repeated updates, state-vector ordering and closed-form checks without assuming prior dynamical-systems experience. What you will practise Iterate a recurrence reliably Identify required initial data Construct a two-component state Distinguish linear and affine updates Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review basic algebra and matrix–vector multiplication. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The final closed-form check requires both the update rule and the initial value to agree. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

linear algebra

Oct 20, 2026

6 questions