Math 304-06: Linear Algebra
Syllabus
Fall 2019
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Syllabus
The textbook is Linear Algebra and Its Applications, 5th Edition, by David C. Lay, Steven R. Lay, and Judi J. McDonald, Pearson Publishers.
This list is what we plan to cover. It will be subject to change as we go, possibly adding or deleting some topics depending on what time allows.
December update: This syllabus is now complete, since the course is over.
- In the assigned sections, omit all "Numerical notes".
- Chapter 1, Sections 1-5, 6-9. Linear systems, matrices and vectors, echelon form. Linear independence and transformations in Rn.
- Chapter 2, Sections 1-3, 8-9. Matrix operations, inversion. Rn as a vector space.
- Chapter 4, Sections 1-7. Abstract and general vector spaces. Linear independence, transformations. Bases, coordinates, and changes thereof.
- Chapter 3, Sections 1, 2, part of 3. Determinants. Volume.
- For § 3: Read about volume and Cramer's Rule, but they will not be on a test.
- Chapter 5, Sections 1-4. Eigenvalues and eigenvectors. Diagonalization. Connection to linear transformations.
- § 1: Omit difference equations.
- Chapter 6, Sections 1-4, 7. Inner product, orthogonality, orthogonal projection. General inner product.
- § 2: Omit decomposing forces.
- § 4: Omit QR factorization.
- Chapter 7, Section 1. Symmetric matrices, their eigenvalues and diagonalization.
- § 2: Omitted due to snow day.
Very Detailed List of Course Contents
This list is not intended to be complete. The complete list is the syllabus, above.
Systems of Linear Equations
Solution by row reduction
Matrices and operations with them
Reduced Row Echelon Form
Sets of Matrices of size mxn
Functions (general theory), injective, surjective, bijective, invertible, composition
Linear Functions ("linear transformations") determined by a matrix
Abstract Vector spaces
Basic theorems, examples, subspaces, linear combinations, span of a set of vectors
Linear transformations L : V → W between vector spaces
Kernel and Range of a linear function
Connection with injective, surjective, bijective, invertible, composition of linear functions, isomorphism
Matrix multiplication from composition, formulas, properties (associativity)
Row (and column) operations achieved by left (right) matrix multiplication by elementary matrices
Theorems about invertibility of a square matrix (if row reduces to the identity matrix), algorithm to compute inverse
General vector spaces; examples including Pn (polynomials), Rm×n (matrices)
Linear independence/dependence, removing redundant vectors from a list keeping span the same
Basis (independent spanning set), Theorems about basis
Dimension of a vector space (or subspace), rank, nullity, theorems about dimension
For L : V → W, dim(V) = dim(Ker(L)) + dim(Range(L)) and its applications
Coordinates as an isomorphism from V to Rn (n×1 matrices)
Representing a linear function L : V → W by a matrix (with respect to choice of basis S of V and basis T of W)
Theorems and algorithms; how matrix representing L changes when bases change to S' and T'
Equivalence of matrices, Block Identity Form
Study special case of L : V → V using same basis S on both ends; linear operators
Effect of change of basis on matrix representing a linear operator, similarity of matrices
Investigate when L might be represented by a diagonal matrix
Eigenvectors, Eigenvalues
Determinants as a tool for finding eigenvalues, general theorems and properties about determinants
det(AB) = det(A) det(B), A invertible iff det(A) not zero
Characteristic polynomial of a matrix, det(A - t I), zeroes ("roots") are eigenvalues
Similar matrices have same characteristic polynomial
Geometric and algebraic multiplicities of eigenvalues for L : V → V (or for matrix A representing L)
Diagonalization of matrices
Theorems (geometric mult less than or equal to algebraic mult); L diagonalizable iff geom mult = alg mult for all eigenvalues
Computational techniques to find a basis of eigenvectors; diagonalization of matrix A, if possible
Geometry in Linear Algebra: dot product, angles and lengths, orthogonality, orthonormal sets, orthogonal projections
Orthogonal matrices
Orthogonal diagonalization of symmetric matrices