Hjem

1. Introduction to Vectors:
1.1 Vectors and linear combinations;
1.2 Lengths and dot products;
1.3 Matrices;
2. Solving Linear Equations:
2.1 Vectors and linear equations;
2.2 The idea of elimination;
2.3 Elimination using matrices;
2.4 Rules for matrix operations;
2.5 Inverse matrices;
2.6 Elimination = factorization: A = LU;
2.7 Transposes and permutations;
3. Vector Spaces and Subspaces:
3.1 Spaces of vectors;
3.2 The nullspace of A: solving Ax = 0;
3.3 The rank and the row reduced form;
3.4 The complete solution to Ax = b;
3.5 Independence, basis and dimension;
3.6 Dimensions of the four subspaces;
4. Orthogonality:
4.1 Orthogonality of the four subspaces;
4.2 Projections;
4.3 Least squares approximations;
4.4 Orthogonal bases and Gram-Schmidt;
5. Determinants:
5.1 The properties of determinants;
5.2 Permutations and cofactors;
5.3 Cramer's rule, inverses, and volumes;
6. Eigenvalues and Eigenvectors:
6.1 Introduction to eigenvalues;
6.2 Diagonalizing a matrix;
6.3 Applications to differential equations;
6.4 Symmetric matrices;
6.5 Positive definite matrices;
6.6 Similar matrices;
6.7 Singular value decomposition (SVD);
7. Linear Transformations:
7.1 The idea of a linear transformation;
7.2 The matrix of a linear transformation;
7.3 Diagonalization and the pseudoinverse;
8. Applications:
8.1 Matrices in engineering;
8.2 Graphs and networks;
8.3 Markov matrices, population, and economics;
8.4 Linear programming;
8.5 Fourier series: linear algebra for functions;
8.6 Linear algebra for statistics and probability;
8.7 Computer graphics;
9. Numerical Linear Algebra:
9.1 Gaussian elimination in practice;
9.2 Norms and condition numbers;
9.3 Iterative methods for linear algebra;
10. Complex Vectors and Matrices:
10.1 Complex numbers;
10.2 Hermitian and unitary matrices;
10.3 The fast Fourier transform; Solutions to selected exercises; Matrix factorizations; Conceptual questions for review; Glossary: a dictionary for linear algebra; Index; Teaching codes.