SyDe114 - Linear Algebra
Lecture Topics and Notes
Lecture Topics and Notes
Stephen
Birkett
|
SyDe114 - Linear Algebra Lecture Topics and Notes |
Stephen
Birkett
|
Research
Teaching
Linear Algebra
Topics and Lectures Assignments Mini-Tests Numerical Methods Computational Mathematics Musical Instruments Music
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The outline and roadmap below may be fine-tuned as we go along, so check back here regularly. This information provides details of specific topics covered and references to the text sections required in addition to the material in the lecture notes. The version at the end of the course will be the definitive one! Links here to pdf files of the lecture slides will become active as these are available for downloading, and updated as we proceed in the lectures. | ||||
TEXT REFERENCES | APPROXIMATE # LECTURES | DETAILED TOPICS | |||
Ch 4; Ch 1.2, 1.3, 1.7, 1.8; Ch 2.1-2.4 | 11 | COURSE INTRODUCTION. | |||
| UNIT 1 - VECTOR SPACES. | |||||
| Definition. Examples: Euclidean spaces, matrices, polynomial spaces, function spaces, binary vector space. Subspaces. More examples. Linear combinations and linear span. Linear independence. Bases and dimension. Rank. Coordinate representation of vectors. Sums and direct sums of vector spaces. | |||||
Ch 2.5-2.12; Ch 3.7, 3.12; Ch 8.2, 8.3, 8.6-8.9 | 4 | UNIT 2 - MATRIX ALGEBRA. | |||
| Matrix multiplication. Transpose. Powers and polynomials of matrices. Elementary row and column operations. Matrix invertibility (non-singular). Finding inverses. Systems of equations. Elementary matrices. Determinants. Diagonal and triangular matrices. Block matrices. | |||||
Ch 5; Ch6.1-6.3 | 8 | UNIT 3 - LINEAR MAPS. | |||
| Definitions: map=mapping=function; transformation; operator. Test for linearity. Special maps. Operations with maps; composition. Matrix transformations. Kernel and image; nullity and rank of a linear map. Non-singular, one-to-one maps, and onto maps and special results applicable for linear maps. Invertible maps. Isomorphism. Matrix representation of a linear operator with respect to an arbitary basis. Change of basis. | |||||
Ch9.1 -9.5; Ch 6.4 Extra Resources: Factoring & Finding Roots | 3 | UNIT 4 - EIGENVALUES, EIGENVECTORS & DIAGONALIZATION. | |||
| Diagonalization. Characteristic polynomial. Eigenvalues and eigenvectors. Eigenspaces. Geometric and algebraic multiplicity of eigenvalues. Diagonalizing matrices. Similarity. | |||||
Ch7.1-7.9; Ch1.4-1.6; Ch2.10; Ch9.6 | 7 | UNIT 5 - INNER PRODUCT SPACES AND ORTHOGONALITY. | |||
| Real and complex inner product spaces. Cauchy-Schwartz inequality. Orthogonality. Equations of (hyper)planes and distance to points in R^n. Cross product in R^3. Orthonormal sets and bases. Orthogonal matrices and orthogonal diagonalization (real symmetric matrices). Quadratic forms and diagonalizing conics. [NOT included: Positive-definite matrices. p-norms.] | |||||
Ch3.1-3.11 | 1 | UNIT 6 - LINEAR SYSTEMS OF EQUATIONS. | |||
| Terminology and definitions. Consistency and uniqueness of solutions. Equivalent systems. Gaussian and Gauss-Jordan elimination. Solution space of a homogeneous system. Non-homogeneous systems. Solution set. Particular and general solution. Geometric interpretation in Euclidean space. Parametric equations of a line in Rn. Under- and over-determined systems. Existence of solutions. | |||||
