A matrix as a move of the whole plane, then the number that says how much it stretches area and volume: its rules, how to compute it, Cramer's rule, the inverse formula, and areas, volumes and cross products. 9 short slides, then worked examples and problems. Each slide has an everyday picture, a worked example and Python you can run right on the page.
Free preview: most of every slide is yours to read, with its Python lab. Members get the full text, every example with its solution, and five extra problems per lesson.
Lesson 6TransformationsA matrix moves the whole planeOpen slide →01 · Strang §8.1 · Linear transformationsSee a matrix move the plane. Why the columns are where (1, 0) and (0, 1) land, how a rotation matrix is built, and what the determinant measures.
Free preview
Lesson 6ReflectionChallenge: build a mirrorOpen slide →02 · ChallengeBuild the 2x2 matrix that mirrors the plane, and see why a reflection has determinant −1. An interactive linear algebra challenge in Python.
Free preview
Lesson 6DeterminantThe determinant: how much a matrix stretches areaOpen slide →03 · Strang §5.1 · What the determinant measuresThe determinant says how much a matrix stretches area (2D) or volume (3D); its sign says whether space is flipped. Examples and Python. Strang §5.1.
Free preview
Lesson 6Three rulesThe three rules that define the determinantOpen slide →04 · Strang §5.1 · The properties of the determinantThree rules define the determinant: det I = 1, a row swap flips the sign, det is linear in each row. All else follows. Examples and Python. Strang §5.1.
Free preview
Lesson 6Det from pivotsComputing the determinant from the pivotsOpen slide →05 · Strang §5.1 · Determinants by eliminationCompute any determinant by elimination: det A = ± (product of the pivots). Also det AB = det A det B and det Aᵀ = det A. Examples and Python. Strang §5.1.
Free preview
Lesson 6CofactorsCofactor expansion: a big determinant from smaller onesOpen slide →06 · Strang §5.2 · Permutations and cofactorsThe big formula and cofactor expansion: a 3×3 determinant from 2×2 pieces with signs + − +. Worked examples and Python. Strang §5.2.
Free preview
Lesson 6Cramer's ruleCramer's rule and the inverse from cofactorsOpen slide →07 · Strang §5.3 · Cramer's rule and inversesCramer's rule solves Ax = b with ratios of determinants, and cofactors give A⁻¹ = Cᵀ / det A. When to use them and when not. Examples and Python. Strang §5.3.
Free preview
Lesson 6Cross productAreas, volumes and the cross product from determinantsOpen slide →08 · Strang §5.3 · Areas, volumes and cross productsDeterminants give a triangle's area from its corners, a box's volume from its edges, and the cross product u × v. Robotics examples and Python. Strang §5.3.
Free preview
Lesson 6RecapWhat you can now do.Open slide →09 · Recap · ReadingLesson 6 on one page: how a matrix moves the plane, the mirror, the determinant as an area and volume factor, the three rules, pivots, cofactors, Cramer's rule, the inverse formula and cross products.
