Syllabus for Math 1564
Linear Algebra with Abstract Vector Spaces
Fall 2015
The timing of topics in this syllabus is approximate
and may change during the semester.
Week 1 (Lectures on 8/18, 8/20):
Text: One.I
Concepts: Standard form of a linear system,
solutions, row operations don't change
solution set, row operations lead to a system
“in row echelon form” whose
solutions are easy to read off. Homogeneous
and inhomogenous systems. Matrices and
vectors, addition and scalar
multiplication. Free variables,
parameterization of the general solution to a
system. The general solution is a particular
solution plus a solution of the associated
homogeneous system. Non-singular square
matrices.
Skills: Given any description of a linear
system, write it in standard form, apply row
operations to transform it to row echelon
form, and read off the solutions. Determine
whether a system has solutions. Give examples
of different outcomes (no solutions, unique
solution, infintely many solutions). Determine
whether a matrix is non-singular.
Text: One.II
Concepts: Vectors as length and
direction. Free vectors and vectors in
standard/canonical position. Geometric
interpretation of addition and scalar
multiplication. Lengths and angles, dot/inner
product, triangle and Cauchy-Schwartz
inequalities.
Skills: Visualize solutions to systems in
two and three variables. Calculate lengths and
angles.
Week 2 (Lectures on 8/25, 8/27):
Text: One.III
Concepts: Reduced echelon
form. Pivoting/Gauss-Jordan reduction leads to
reduced echelon form. Solutions can be read
off easily from this form. Equivalence
relations and equivalence classes; row
equivalence of matrices. Linear combinations
and row operations. Each row equivalence class
contains a unique matrix in reduced echelon
form.
Skills: Carry out Gauss-Jordan reduction and
interpret the results in terms of solutions of
linear systems.
Text: One.Topic(Accuracy of Computations)
Concepts: Roundoff errors can be amplified by
poor choices in reduction
algorithm. “Partial pivoting” is one
approach to limiting errors.
Skills: Row reduction with partial
pivoting.
Week 3 (Lectures on 9/1, 9/3):
Text: Two.I.1
Concepts: Motivation for abstract vector
spaces as the right framework for studying
linear combinations. Definition and first
examples of vector spaces. Additional
“rules of calculation” that follow
from the axioms.
Skills: Give examples and non-examples of
vector spaces.
Text: Two.I.2
Concepts: Definition and examples of
subspaces. Span of a set of vectors.
Subspaces are spans.
Skills: Give examples of subspaces. Check
whether a vector is in the span of a set of
vectors.
Week 4 (Lectures on 9/8, 9/10):
Text: Two.II
Concepts: Motivation via minimal spanning
sets. Definition of linear independence and
equivalent conditions. Relationship between
span and linear independence.
Skills: Check whether a set of vectors is
linearly independent.
Text: Two.III.1
Concepts: Definition of a basis.
Standard basis of Rn. Other
examples of bases. Coordinates of a vector
with respect to a basis.
Skills: Check whether a set of vectors is
a basis. Compute the coordinates of a vector
with respect to a basis.
Text: Two.III.2
Concepts: Finite-dimensional vector
spaces. All bases have the same number of
elements. Dimension is the number of elements
in a basis. Bases obtained by adding to
a linear independent set or removing from a
spanning set.
Skills: Compute the dimension of a vector space.
Week 5 (Lectures on 9/15, 9/17):
Text: Two.III.3
Concepts: Row span, row rank, column
span, and column rank of a matrix. Equality
of row rank and column rank. Rank and the
dimension of the set of solutions of a
homogeneous linear system.
Skills: Compute the rank of a matrix.
Compute the dimension of the set of solutions
of a homogeneous linear system.
Exam 1 on 9/22 covering material
discussed up to and including Week 4
Week 6 (Lecture on 9/24):
Text: Three.I
Concepts: Definition of an isomorphism of
vector spaces and basic examples.
Automorphisms. Isomorphisms preserve linear
independence and bases. Isomorphism is an
equivalence relation. Finite-dimensional
vector spaces are isomorphic if and only if
they have the same dimension.
Skills: Check whether a map is an
isomorphism. Give interesting examples of
isomorphisms between standard spaces
(Rn, polynomials, ...)
Week 7 (Lectures on 9/29, 10/1):
Text: Three.II
Concepts: Definition, basic properties,
examples of linear maps. Set of linear maps
between two vector spaces is itself a vector
space. Image (=range), kernel (=nullspace) of a
linear map. Dimension formula. Non-singular
(=injective=1-1) linear maps.
Skills: Check whether a map is linear.
Give examples and non-examples of linear maps
between standard spaces. Compute the image
and kernel of a linear map and their
dimensions. Determine whether a linear map is
non-singular.
Week 8 (Lectures on 10/6, 10/8):
Text: This material is implicit in Three.IV
Concepts: Linear maps from Rn to Rm
are given by matrices. Composition corresponds to matrix multiplication.
Examples of interesting maps/matrices: dilations, rotations, shears,
projections.
Skills: Compute matrix-vector products, find the matrix representing a
map given geometrically. Compute properties of a map (rank, nullity, kernel, image)
from the matrix.
Text: Three.III
Concepts: Coordinates of a linear map with
respect to two bases. Composition and matrix multiplication.
Skills: Compute coordinates of linear
maps. Find properties of a linear map (rank, nullity, kernel, image)
from its coordinate matrix.
(Homework this week due 10/15)
Week 9 (Lecture 10/15):
Text: Three.IV
Concepts: Special matrices: zero, identity, permutation matrices,
elementary matrices. Row reduction via matrix multiplication.
Calculation of inverse matrices.
Skills: Identify special matrices. Calculate inverses.
Week 10 (Lectures on 10/20, 10/22):
Text: Three.V
Concepts: Effect of changing bases on coordinates of vectors and matrices
of linear maps. Change of basis formulas. Standard form for a linear map.
Skills: Compute change of basis matrices and use them appropriately in change
of basis formulas.
Week 11 (Lectures on 10/27, 10/29):
Text: Four.I.1-2, Four.II, Four.III.1
Concepts: Determinants detect non-singularity. They can be calculated
using row operations. Determinants as volumes. Laplace expansion.
Skills: Calculate determinants using row operations and the Laplace
expansion.
Exam 2 on 11/3 covering material discussed up to and including Week 10
Week 12 (Lecture on 11/5):
Text: Three.VI.1-2
Concepts: The dot product as a symmetric, bilinear, positive definite function of two vectors. Orthogonality. Orthogonal projection of a vector onto the line spanned by another vector. Orthogonal and orthonormal sets and bases. The Gram-Schmidt algorithm.
Skills: Compute and interpret dot products. Compute projections. Apply the Gram-Schmidt algorithm.
Week 13 (Lectures on 11/10, 11/12):
Text: Notes to be distributed
Concepts: Coordinates with respect to an orthonormal basis. Orthogonal projection onto a subspace. Best approximation of a vector by another vector constrained to lie in a subspace. Applications to inconsistent and/or underdetermined systems. Application to least squares approximation.
Skills: Compute orthogonal projections and best approximations. Compute least squares approximations.
Week 14 (Lectures on 11/17, 11/19):
Text: Five.I, Five.II
Concepts: Complex vector spaces. Similarity of matrices. Eigenvalues and eigenvectors. Characteristic polynomial. Diagonalization and bases of eigenvectors.
Skills: Compute characteristic polynomials, eigenvalues, and eigenvectors. Use the change of basis formula to diagonalize a matrix.
Week 15 (Lecture on 11/24):
Text: Class notes and Five.IV.2.13
Concepts: Jordan blocks and Jordan matrices. Every linear transformation is represented by an essentially unique Jordan matrix. Relationship between characteristic polynomial and Jordan form.
Skills: Compute the Jordan form directly in small dimensions. Deduce the Jordan form from information on eigenvalues and kernels.
Week 16 (Lectures on 12/1, 12/3):
Text: Five.Topic(Method of Powers), Five.Topic(Page Ranking)
Concepts: Computing eigenvalues andd eigenvectors via iteration of a map. Applying eigenvalues and eigenvectors to ranking web pages.
Concepts: Overview of course. A few topics not covered in the course that you should be aware of.
Final Exam on Tuesday December 8th from 2:50pm to
5:40pm covering material up to and including week 15
Additional topics that may be covered if time allows:
Text: One.Topic(Networks)
Concepts: Kirchhoff's laws.
Skills: Set up a linear system describing
the currents in a DC circuit. Solve the system
and interpret the results.
Text: Three.Topic(Markov Chains) + notes
Concepts: Definition and examples of
Markov chains
Skills: Set up a matrix modeling a Markov
chain and use it to explore its long-term
behavior