Subspaces, the nullspace, pivot and free columns, the complete solution to Ax = b, independence and dimension, and the big picture of the four subspaces. 7 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 4SubspacesSubspaces: lines and planes through the originOpen slide →01 · Strang §3.1 · Spaces of vectorsA subspace is a set you cannot leave by adding and scaling: a line or plane through the origin. Examples, non-examples and live Python. Strang §3.1.
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Lesson 4NullspaceThe nullspace: finding the inputs that give zeroOpen slide →02 · Strang §3.2 · The nullspace of AThe null space is every x with Ax = 0: inputs that produce no output. A walking picture, worked 2×2 and 3×3 examples and interactive Python. Strang §3.2.
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Lesson 4Special solutionsFinding the nullspace: reduce to R, then read off the special solutionsOpen slide →03 · Strang §3.2 · Solving Ax = 0 and Rx = 0Reduce A to R, spot the pivot and free columns, and build the special solutions that fill the null space. Worked 2×4 and 3×4 examples with Python. Strang §3.2.
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Lesson 4Complete solutionSolving Ax = b completely: one solution plus the nullspaceOpen slide →04 · Strang §3.3 · The complete solution to Ax = bEvery solution of Ax = b is one particular solution plus any null space vector. See the solution line in an interactive plot and find it in Python. Strang §3.3.
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Lesson 4Basis and dimensionIndependence, basis and dimension: how many directions a space needsOpen slide →05 · Strang §3.4 · Independence, basis and dimensionIndependent vectors, a basis and the dimension of a space, with a walking-directions picture, a 3D example and rank in Python. Strang §3.4.
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Lesson 4Four subspacesThe four fundamental subspaces and their sizesOpen slide →06 · Strang §3.5 · Dimensions of the four subspacesRow space, column space, null space and left null space: their dimensions and how they fit together, with an interactive picture. Strang §3.5.
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Lesson 4RecapWhat you can now do.Open slide →07 · Recap · ReadingChapter 3 on one page: subspaces, the null space, special solutions, the complete solution to Ax = b, basis, dimension and the four subspaces.
