MATH 111 · Year 1 · Semester 1 · 3 credits · Mathematics & Statistics

Linear Algebra

The geometric grammar of modern risk: transforming data, Markov chains, and predictive models through vector spaces and spectral theory.

Full practice exam · 30 Q · 120 min

Season 1 · 8 episodes

The Architecture of Invariance: Linear Algebra at Apex Mutual

A newly appointed quantitative risk analyst must rebuild an insurer's capital and reserving models from first principles using linear algebra before a regulatory audit shuts down their flagship annuity and health lines.

Protagonist · Maya Lin, an analytical and determined junior quantitative risk analyst at Apex Mutual Assurance.

Setting · The actuarial risk and ALM floor at Apex Mutual Assurance, overlooking the City of London financial district.

Stakes · A four-hundred-million-pound solvency capital deficit looms ahead of a binding Prudential Regulation Authority audit, threatening policyholder security and the firm's operating licence.

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Episodes

Season finale: Maya unifies multi-state life reserving, bonus-malus transitions, and dynamic hedging into a single coherent linear-algebraic framework, proving solvency and presenting the model to the regulator.

From the lecturer: Welcome to Linear Algebra. To master this course, do not treat it as a spectator sport. Keep a grid notebook and pen beside you during every lecture film: whenever a matrix reduction or geometric transformation appears, pause the video and execute the calculation or sketch the row and column spaces before watching my reveal. Linear algebra sits at the absolute foundation of your actuarial and data science career. While not tested under a standalone preliminary exam code, its language drives SOA Exam SRM (Statistics for Risk Modeling), CAS Exams MAS-I and MAS-II (generalized linear models and time series), and the multi-state transition frameworks in advanced life contingencies. When you see a linear system, train your brain to hold two views simultaneously: the algebraic mechanics of row operations and the clean geometry of vector subspaces. We will derive every core result rigorously from first principles—from Gaussian elimination to the Singular Value Decomposition—grounding each abstraction in portfolio risk, Markov transition matrices, and normal equations for predictive modeling. Let us begin building your mathematical foundation.

Cold open

At six forty-five in the morning, Maya stares at a cash-flow mismatch report. Apex has ninety million pounds of guaranteed annuity liabilities maturing across the next three years, but the trading desk's bond portfolio leaves two million unfunded in year two while locking up excess capital in year three.

preparing lecture…

Transcript

Formulate systems of linear equations in augmented matrix notation for actuarial and financial models.
Perform the three elementary row operations systematically to achieve Row Echelon Form (REF) and Reduced Row Echelon Form (RREF).
Classify linear systems as consistent (unique or infinite solutions) or inconsistent using pivots and rank.
Evaluate the computational efficiency and numerical stability of Gaussian elimination with partial pivoting.