🧮 Linear Algebra

What you’ll learn

1. The Math Behind the Modern WorldMotivate linear algebra as the computational language of images, robotics, and data.Vectors (lists of numbers) and matrices (grids of numbers) are how computers represent images, forces, models, and measurements. Robots, graphics engines, GPS, and search all reduce to matrix arithmetic, which is why this course's ideas power most modern technology.
2. Vectors: Arrows and ListsDefine vectors geometrically and numerically, with magnitude, components, unit vectors, and dimension.A vector is simultaneously an arrow (length + direction) and a coordinate list, and physics' forces and velocities are its natural examples. Magnitude comes from the Pythagorean theorem, and nothing changes conceptually as dimension climbs from 2 to 50.
3. Vector ArithmeticPerform vector addition and scalar multiplication, and connect them to real navigation problems.Vectors add tip-to-tail, which is just componentwise addition; scalars rescale and can flip direction. Crosswind landings and river crossings are literal vector sums solved continuously by pilots and autopilots.
4. The Dot ProductCompute dot products and interpret them via |a||b|cos θ, especially the orthogonal case.The dot product multiplies matching components and adds, equaling |a||b|cos θ: positive means agreement, zero perpendicularity, negative opposition. Solar-panel orientation and document-similarity scoring are direct applications.
5. Linear Systems: Where Lines MeetFrame simultaneous linear equations geometrically and historically.A linear system asks for the intersection of lines (or planes), and the Chinese 'Nine Chapters' was solving such systems with rod arrays two millennia ago — an early matrix method. Systems can have one solution, none, or infinitely many.
6. Gaussian EliminationExecute Gaussian elimination using the three legal row operations.Elimination reduces a system to triangular form via row swaps, scalings, and row additions, then back-substitutes — the same algorithm, industrialized, that solves million-equation engineering models today. Gauss systematized it; ancient China anticipated it.
7. Matrices: Data in RectanglesDefine matrices, their shapes, and the special identity, diagonal, and zero matrices.A matrix is an m×n table of numbers — census cards, spreadsheets, and images are all matrices — and Cayley's 1858 memoir turned such tables into algebraic objects. The identity acts like 1, the zero matrix like 0, and diagonal matrices stretch each axis independently.
8. Matrix MultiplicationMultiply matrices by the row-times-column rule and understand multiplication as composition.Entry (i, j) of AB is row i of A dotted with column j of B, sized (m×n)(n×p) = (m×p). The rule exists so that AB means 'do B, then A' — composing transformations like assembly-line stages — and order genuinely matters: AB ≠ BA in general.
9. Matrices as TransformationsRead 2×2 matrices as geometric transformations of the plane.A matrix moves every vector at once — rotating, scaling, reflecting, or shearing — while keeping lines straight and the origin fixed. Its columns are the landing spots of the basis vectors, so watching the unit square reveals the whole transformation.
10. The DeterminantCompute 2×2 determinants and interpret them as area scale factors and invertibility tests.det[[a,b],[c,d]] = ad − bc measures how the transformation scales area, with sign recording orientation flips. Seki Takakazu (1683) and Leibniz discovered determinants independently, and det = 0 signals collapse: no inverse, no unique solution.
11. Inverses: Undoing a MatrixFind 2×2 inverses and use them conceptually to solve Ax = b.The inverse undoes a matrix (AA⁻¹ = I) and exists exactly when det ≠ 0; for 2×2, swap the diagonal, negate the off-diagonal, divide by det. Ciphers depend on invertible operations — Hill's 1929 cipher decrypts with a matrix inverse — and x = A⁻¹b explains when systems have unique answers.
12. Eigenvalues and EigenvectorsDefine eigenvectors and eigenvalues and connect them to structural vibration modes.Eigenvectors are the directions a matrix only stretches (Av = λv), λ being the stretch factor, findable for simple matrices by inspection. Vibration modes of bridges and buildings are eigenvectors of their governing equations — the 1940 Tacoma Narrows collapse is engineering's canonical cautionary tale.
13. Eigen-Applications: PageRank and SkyscrapersApply eigen-thinking to PageRank and tuned mass dampers.PageRank is the dominant eigenvector of the web's link matrix, computed by power iteration — repeatedly multiplying until scores stabilize. The same mathematics, run in reverse as design, tunes Taipei 101's 660-tonne damper to the tower's natural frequency.
14. Least Squares: Fitting DataUse least squares to reconcile overdetermined, noisy systems and fit trend lines.When there are more equations than unknowns, least squares minimizes total squared error — the method Gauss used to recover the asteroid Ceres in 1801. Regression lines, GPS fixes, radio interferometry, and model training are all the same idea at different scales.
15. Where Linear Algebra Takes YouConsolidate the course and map its concepts onto modern computing careers.GPUs are matrix-multiplication engines built for graphics that now train neural networks, whose layers are matrix products. Transformations, systems, eigenvectors, least squares, and dot products each anchor entire industries — and you now know all five.

Questions this course answers

Grounded in trusted sources

• Gilbert Strang, 'Introduction to Linear Algebra', 5th ed. (Wellesley-Cambridge Press, 2016)
• David C. Lay et al., 'Linear Algebra and Its Applications', 5th ed. (Pearson, 2016)
• Kangshen Shen et al., 'The Nine Chapters on the Mathematical Art' (Oxford University Press, 1999)
• Arthur Cayley, 'A Memoir on the Theory of Matrices', Philosophical Transactions of the Royal Society (1858)
• Brin & Page, 'The Anatomy of a Large-Scale Hypertextual Web Search Engine', WWW7 (1998)
• Victor Katz, 'A History of Mathematics', 3rd ed. (Pearson, 2008)
• Teets & Whitehead, 'The Discovery of Ceres: How Gauss Became Famous', Mathematics Magazine 72 (1999)

Every Wunder lesson is built from real, reputable sources — never invented.

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