SyDe114  Linear Algebra Lecture Topics and Notes 
Stephen
Birkett

Research
Teaching
Linear Algebra
Topics and Lectures Assignments MiniTests Numerical Methods Computational Mathematics Musical Instruments Music

The outline and roadmap below may be finetuned 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 2.12.4  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 2.52.12; Ch 3.7, 3.12; Ch 8.2, 8.3, 8.68.9  4  UNIT 2  MATRIX ALGEBRA.  
Matrix multiplication. Transpose. Powers and polynomials of matrices. Elementary row and column operations. Matrix invertibility (nonsingular). Finding inverses. Systems of equations. Elementary matrices. Determinants. Diagonal and triangular matrices. Block matrices.  
Ch 5; Ch6.16.3  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. Nonsingular, onetoone 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.  
Ch7.17.9; Ch1.41.6; Ch2.10; Ch9.6  7  UNIT 5  INNER PRODUCT SPACES AND ORTHOGONALITY.  
Real and complex inner product spaces. CauchySchwartz 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: Positivedefinite matrices. pnorms.]  
Ch3.13.11  1  UNIT 6  LINEAR SYSTEMS OF EQUATIONS.  
Terminology and definitions. Consistency and uniqueness of solutions. Equivalent systems. Gaussian and GaussJordan elimination. Solution space of a homogeneous system. Nonhomogeneous systems. Solution set. Particular and general solution. Geometric interpretation in Euclidean space. Parametric equations of a line in Rn. Under and overdetermined systems. Existence of solutions.  