Math 214 Spring 2007: Lecture Notes

Math 214 Linear Systems Spring 2007	Schedule	Students	Resources	Projects	Ron Buckmire: Contact & Home & Schedule
Math 214 Linear Systems Spring 2007	Syllabus	Quizzes	Exams	HW &Notes	Ron Buckmire: Contact & Home & Schedule

Math 214 Spring 2006: Lecture Notes

#	Date	Topic	Notes
Class 31	Mon Apr 16	Review for Exam 2	Think of all the cool topics we have learned since the last exam!
Class 30	Fri Apr 13	Least Squares Approximation	We know there are situations where Ax=b has no solution (i.e. b is not in the col(A)). Finding the projection of b onto col(A) allows us to find "the next best" solution to Ax=b, often called the least squares approximation. This involves solving A^TA x = A^Tb instead. Tbese are known as the normal equations. What is going on is that you are finding the x vector which solves the projection of b into the column space of A, instead of b itself.
Class 29	Wed Apr 11	Projection Matrices	We can use some tricky algebra to get an expression for how to project a vector on to the columns space of a matrix A.
Class 28	Mon Apr 9	Gram-Schmidt Orthogonalization	Suppose you have a basis for a subspace and you want to swap that for an orthogonal basis, what do you do? Use the Gram-Schmidt process. Then you can use your orthogonal basis to project a vector NOT in the subspace onto the subspace by adding up the projections onto each of the orthogonal basis directions. I like to think of this as adding up all the individual primary colors which make up a standard color.
Class 27	Fri Apr 6	Orthogonal Complements.	The deep relationships between the associated subspaces of a matrix are revealed and the concept of orthogonal complement introduced.
Class 26	Wed Apr 4	Orthonormal Matrices.	Professor Mickey McDonald from Linear Systems Fall 2006 wrote up content goals for Section 5.1, 5.2, 5.3 and 5.4.
Class 25	Mon Apr 2	Numerical Computation of Eigenvalues	We shall use Matlab to explore how eigenvalues are often computed in real life, which is an algorithm known as the power method. Content Goals for Chapter 4.
Class 24	Fri Mar 30	Similarity and Diagonalization.	We shall begin to explore diagonalizability and matrix exponentiation. Introduction of the concept of an equivalence relation (satisfies symmetry, reflexivity and transitivity). Recall that diagonalizability is just one instance of matrices being similar to each other, i.e. if P exists then AP=PB means A ~B. Think of P coming between the A and B. For diaginalizability, P is a matrix where the columns are the eigenvectors of A. The geometric and algebrtaic multiplicities have to be the same for a matrix to be diagonalizable.
Class 23	Wed Mar 28	Eigenvalues and Eigenvectors of nxn Matrices	We will expand our ability to compute eigenvalues and eigenvectors and eigenspaces to all square nxn matrices
Class 22	Mon Mar 26	Determinants	Computing determinants involves understanding a recursive (self-referential) definition, like the definition of factorial, or n! We shall also learn about important interpretations of determinant as well as applications such as the vector cross product and Cramer's Rule.
Class 21	Fri Mar 23	Introduction to Eigenvalues and Eigenvectors	We introduce the concepts of eigenvalues and eigenvectors and eigenspaces first in the context of 2x2 matrices. There are some lovely graphical representations and interpretations of these concepts.
Class 20	Wed Mar 21	Applications of Linear Algebra: Graph Theory	This a cool little topic which introduces you to the topic of graph theory and shows you a taste of how matrices can be used to answer questions in that subject. Specifically we will learn about adjacency matrices. Take Math 382 Graph Theory next year for more information on this subject!
Class 19	Mon Mar 19	Linear Transformations	We introduce another way of looking at the Ax=b problem, that is as a linear transformation by a matrix A of a vector x into a new vector b. This has a nice visual representation and applications to a number of other subjects in mathematics.
Class 18	Fri Mar 9	Subspaces Associated With Matrices (continued)	We will introduce the concept of null space and basis (a linearly independent span) and formally define dimension. We will also discuss coordinates. We did examples to show how to compute null(A), row(A) and col(A) from rref(A).
Class 17	Wed Mar 7	Subspaces Associated With Matrices	We will specifically talk about the row space and column space and connect them to the concept of span. We did examples of proving a given space is a subspce or not (Does it contain 0? Is it closed under vector addition? Is it closed under scalar multiplication?)
Class 16	Mon Mar 5	Vector Spaces and Subspaces	An introduction to the most fundamental concept in the course: vector space, along with the associated idea subspace.
	Fri Mar 2	Exam 1
Class 15	Wed Feb 28	Exam 1 Review	A review of the central ideas in the course so far. We can also go over the Practice Exam. Think about how the concepts of span, linear independence, rank, appearance of rref(A) and singularity of A all interrelate with each other.
Class 14	Mon Feb 26	LU Decomposition and Permutation Matrices	Introduction of the first matrix product formula (i.e. matrix factorization) in the course. Also another form of elementary matrix, the permutation matrix is introduced.
Class 13	Fri Feb 23	Elementary Matrices	A new way of thinking of row reduction (and Gauss-Jordan elimination) as a series of matrix multiplications with elementary matrices. Introduction of the first form of the Fundamental Theorem of Invertible Matrices, a central theme of the Poole textbook.
Class 12	Wed Feb 21	The Matrix Inverse	Introduction of the "inverse under matrix multiplication." Usually just called the inverse matrix. Formalization of the Gauss-Jordan elimination and revelation of its significance.
Class 11	Fri Feb 16	Matrix Algebraic Operations	Reinforcement of the concepts of linear independence and span and formal definition of the arithmetic operations on matrices (scalar multiplication and matrix multiplication) and their properties. Introduced idea of linear independence and span being applied to any similarly formally defined objects with associated operations.
Class 10	Wed Feb 14	Matrix Properties	Introduction to the basic algebraic properties of matrices. Some time spent on the definition of "additive identity" and "multiplicative identity" under matrix addition and matrix multiplication as well as real number addition and multiplication. First taste of block matrices.
Class 9	Mon Feb 12	Applications of Linear Systems	The important idea here were the concepts of linear independence and linear dependence. Specifically, the result from Theorem 2.8 that a collection of m vectors in Rⁿ is linearly dependent, if m > n. In addition, the definition of span as the set which consists of ALL possible linear combinations of a given set of vectors.
Class 8	Fri Feb 9	Linear Independence and Span	We discuss examples of undetermined systems and relate these to coefficient matrices with more columns than rows (ie. n>m). Continue discussion of the Rank Theorem and the notion of free variables.
Class 7	Wed Feb 7	Reduced Row Echelon Form and Rank	This class has examples of row reduction, the introduction of reduced row echelon form, rref(A), and rank(A). Begin discussing the Rank Theorem.
Class 6	Mon Feb 5	Linear Systems of Equations	In this class we introduce the concepts of equivalent systems, elementary row operations and row reduction.
Class 5	Fri Feb 2	Understanding Linear Systems of Equations	This class involves thinking about different interpretations of "the linear systems" problem of simultaneous linear equations from a geometric perspective (as the intersection of physical objects) and algebraically in three different ways (linear combination problem, simultaneous linear equations and matrix linear transformations involving matrix multiplication). The important result is that all linear systems have either 0 or 1 or an infinite number of solutions.
Class 4	Mon Jan 29	Equations of Planes	Algebraic geometry with planes: parametric, vector and normal form. In R³ the important thing to remember is that the normal vector defines a plane uniquely. An important result here is The Number of Equations it takes to describe an object plus the Dimension of the Object Equals the Dimension of the Space it lies in.
Class 3	Fri Jan 26	Projections and Equations of Lines	Thinking about a familiar topic (equations of lines) in an unfamiliar setting (using vectors) prepares you for analytic geometry in higher dimensions.
Class 2	Wed Jan 24	Length and Dot Products	Definition of Linear combination. Visualization of same. Understanding the correspondence between the linear combination question and the solution of a system of linear equations. The dot product is defined and is a scalar.
Class 1	Mon Jan 22	Scalars and Vectors	Vectors have a starting point and an ending point. Vector have magnitude and direction. "Standard position" is the origin as starting point. Vector addition (and scalar multiplication) can be visualized. Vector multiplication is not defined.

Last Modified 16 April 2007