1
Eigenvalues and eigenvectors
6 lessonsA vector that a transformation only stretches, together with the number by which it stretches.
1.2The characteristic equation and finding eigenvaluesLinear Algebra
1.3Complex eigenvalues and rotation-scaling matricesLinear Algebra
1.4Finding an eigenvectorLinear Algebra
1.5DiagonalizationLinear Algebra
1.6Matrix powers, dynamical systems and steady statesLinear Algebra
2
Matrices
13 lessonsRectangular arrays of numbers, with their own arithmetic.
2.1Adding, subtracting and scaling matricesPrecalculus
2.2Computing a determinantPrecalculus
2.3Determinants by cofactor expansionLinear Algebra
2.4Matrices
2.5Matrix multiplicationPrecalculus
2.6Rows, columns and dimensionsPrecalculus
2.7The transpose and symmetric matricesLinear Algebra
2.8What the determinant meansLinear Algebra
2.9Writing a system as a matrix equationPrecalculus
2.10Inverse of a 2 by 2 matrix, and solving with itPrecalculus
2.11Properties of determinants: row operations, products, transposesLinear Algebra
2.12Finding an inverse by row reducing A next to ILinear Algebra
2.13Cramer's rulePrecalculus
3
Orthogonality and least squares
4 lessonsPerpendicular vectors, projection onto a subspace, and the best approximate solution of a system that has no exact one.
3.1Orthogonal bases and projection onto a subspaceLinear Algebra
3.3Least squares: the best approximate solution of Ax = bLinear Algebra
3.4The Gram-Schmidt processLinear Algebra
4
Vectors in n dimensions
9 lessonsLists of numbers with addition and scaling; linear combinations and span.
4.1Adding and scaling vectors in n dimensionsLinear Algebra
4.2Linear combinationLinear Algebra
4.3Linear independenceLinear Algebra
4.4SpanLinear Algebra
4.5SubspacesLinear Algebra
4.6Basis and dimensionLinear Algebra
4.7Coordinates in a basis and change of basisLinear Algebra
4.8Dot product, length and orthogonality in n dimensionsLinear Algebra
5
Matrices as linear transformations
10 lessonsA matrix moves every vector in the plane; columns show where the basis goes.
5.1Composing transformations as a matrix productLinear Algebra
5.2Finding the matrix of a transformationLinear Algebra
5.3ImageLinear Algebra
5.4Linear transformationLinear Algebra
5.5Multiplying a matrix by a vectorLinear Algebra
5.6KernelLinear Algebra
5.7Rank and nullityLinear Algebra
5.8When a matrix or transformation can be undoneLinear Algebra
5.9The invertible matrix theoremLinear Algebra
6
Row reduction
8 lessonsSystematic elimination on a matrix to solve any linear system.
6.1Augmented matrices and row operationsPrecalculus
6.2Echelon forms and Gaussian eliminationPrecalculus
6.3Reduced row echelon formLinear Algebra
6.5Homogeneous systems and solutions in parametric vector formLinear Algebra
6.6RankLinear Algebra
6.7Linear systems in use: networks, chemical equations, fitting curvesLinear Algebra