Important dates and times: class meets MW 4:45-6:00 pm, Phys 157
First class: W September 5, 2007
Exam 1: M October 15, 2007
Exam 2: W November 19, 2007
Exam 3: W December 12, 2007
Last class: W December 12, 2007
You may check your homework assignments here
Lecture 1: W Sep 5, 2007
Solved several systems of linear equations and commented on the number of solutions they might have (0, 1 or an infinite
family.) We introduced the elementary row transformations and noticed that they do not change the solution(s) of the system. A first
example of using the elimination algorithm.
This covers mostly Section 1.1 in the book.
Lecture 2: M Sep 10, 2007
We solved another system using the Elimination
Algorithm. We defined matrices (square, upper- and
lower-triangular, diagonal) and vectors. Standard
representation of a vector. The coefficient matrix
and the augmented matrix associated to a system of
linear equations. The reduced row-echelon form of a
matrix.
This covers Section 1.2 in the book.
Lecture 3: W Sep 12, 2007
HW 1 due
We discussed on the number of solutions of a system of linear equations in
terms of leading variables and free variables. Rank of a matrix. The Kronecker-Capelli theorem. Matrix algebra. Linear
combinations of vectors. The matrix form of a linear system. This covers most
of Section 1.3 in the book.
Lecture 4: M Sep 17, 2007
HW 2 due
More on matrix algebra: A(x+y)=Ax+Ay and A(kx)=k Ax. Linear transformations.
The matrix of a linear transformation T:R^m->R^n is given by the values of T
at the standard basis' elements. Equivalent definitions for linear transformations.
Geometric interpretation. This is material from Section 2.1 in the book.
Lecture 5: W Sep 19, 2007
We discussed several geometric transformations which are linear
transformations of the plane (or 3 dim'l space) into itself. Scalings,
projections and reflections over a given line in the plane and in 3-dim'l
space, rotations. This is material from Section 2.2. Read
at home about sheers.
Lecture 6: M Sep 24, 2007
HW 3 due
We talked about invertible linear transformations and invertible matrices. I presented a criterion for determining if a matrice is invertible or not and an
algorithm for finding the inverse matrix. This is material from Section 2.1 and Section 2.3 in the book.
Lecture 7: W Sep 26, 2007
We defined the product of 2 matrices as the matrix associated to the composition of the corresponding linear transformations. We computed the inverse of the
product of two invertible matrices and also studied various properties as associativity and distributivity. Partitioned matrices. This covers the material
from Section 2.4 in the book.
Lecture 8: M Oct 1, 2007
HW 4 due
The image and the kernel of a linear transformation/matrix were defined today. Examples of computing generators for these. The linear span of a set of vectors and
linear subspaces in R^n; first examples (Ker(T) and Im(T)). Characterization of square matrices with trivial kernel. This covers Section 3.1. and some part
of Section 3.2. in the book.
Lecture 9: W Oct 3, 2007
More on linear subspaces. Examples. The subspaces of R^2 and R^3. Relations among vectors. Linearly independent vectors and characterizations. Basis of a
spaces. This finishes Section 3.2 in the book.
Lecture 10: M Oct 8, 2007
HW 5 due
We defined the dimension of a linear subspace in R^n. A basis is a set of linearly independent vectors with maximal number of elements.
A basis is a spanning set with minimal number of elements. Algorithms for computing a basis for the kernel and the image of a matrix.
This covers Section 3.3. in the book.
Lecture 11: W Oct 10, 2007
Review for the first exam.
Lecture 12: M Oct 15, 2007
EXAM 1
Lecture 13: W Oct 17, 2007
Coordinates of a vector with respect to a given basis. The matrix of a linear transformation with respect to a given basis. Similar
matrices. This covers Section 3.4 in the book.
Lecture 14: M Oct 22, 2007
HW 6 due
We talked about linear spaces, and how much of the linear algebra from R^n can be extended to linear spaces. Linear combinations, span,
linearly independent vectors basis and dimension. Examples. Coordinate transforms. This covers material from Section 4.1 in the book.
Lecture 15: W Oct 24, 2007
Linear transformations and associated subspaces: the kernel and the image.
Rank and nullity. Isomorphic linear spaces. The matrix of a linear
transformation and how it varies when we change the basis. This covers material from Sections 4.2 and 4.3 in the book.
Lecture 16: M Oct 29, 2007
HW 7 due
The dot product. Properties. Perpendicular vectors. Orthonormal bases and proof of their existence. The orthogonal complement and its properties. These are topics from Section 5.1.
Lecture 17: W Oct 31, 2007
Orthogonal projection onto a subspace. The Gram-Schmidt algorithm for computing an orthonormal basis for a subspace. The QR-factorization of matrices. These are
topics from Section 5.1 and Section 5.2.
Lecture 18: M Nov 5, 2007
HW 8 due
Pythagoras Theorem in R^n. The Cauchy-Schwarz inequality and applications. The angle of two non-zero vectors. Orthogonal transformations and matrices. Examples and
equivalent definitions. The transpose of a matrix. The dot product viewed as the product of a row vector and a column vector. These are some topics from Sections 5.1,
5.2 and 5.3.
Lecture 19: W Nov 7, 2007
Determinants. Minors and cofactors. Laplace expansion of the determinant. Examples. Rules for computing determinants for 2x2 and 3x3 matrices. Sarrus' rule. The determinant
of a lower/upper triangular matrix. How to compute the determinant using elementary row transformations. These are topics from Sections 6.1 and 6.2 in the book.
Lecture 20: M Nov 12, 2007
HW 9 due
More about determinants. Using induction for computinging determinants.The adjoint matrix. This is material from Chapter 6.
Lecture 21 : W Nov 14, 2007
Connections with the QR decomposition. The determinant of orthogonal matrices. The area of a parallelogram and the volume of a parallelipiped as determinants. Material from Section 6.3 in the
book. Review for the exam on Monday.
Lecture 22 : M Nov 19, 2007
EXAM 2
Lecture 23: W Nov 21, 2007
Least squares approximation method. Data fitting. Spaces with an inner product. Fourier analysis. This is material from Sections 5.4 and
5.5 in the book.
Happy Thanksgiving!
Lecture 24 : M Nov 26, 2007
Eigenvalues and eigenvectors. The characteristic polynomial of a matrix. Compute eigenvalues and eigenvectors. These are some topics from Sections 7.1, 7.2 and 7.3
in the book.
Lecture 25 : W Nov 28, 2007
HW 10 due
More on the characteristic polynomial. The case of an upper triangular matrix. The algebraic multiplicity of an
eigenvalue. Trace of a matrix. Eigenspaces and geometric multiplicities. Geometric versus algebraic multiplicity. This is
material from Sections 7.2 and 7.3 in the book.
Lecture 26: M Dec 3, 2007
Eigenbases and diagonalization of matrices. Linear transformations whose matrix is diagonalizable. Finding
an eigenbasis. Orthogonally diagonalizable matrices. Finding an orthonormal eigenbasis (via the Gram-Schmidt
algorithm). The spectral theorem : characterization of orthogonally diagonalizable matrices. Similar
matrices and some invariants they share: characteristic polynomials, eigenvalues, multiplicities, det, Tr.
Computing the powers of diagonalizable matrices. This is material from Sections 7.3, 7.4 and 8.1 in the book.
Lecture 27: W Dec 5, 2007
HW 11 due
Discrete dynamical systems. The coyote-roadrunner case. The long term evolution of the system depending on the initial
data. Examples. The "counting of rabbits" problem of Fibbonaci and its mathematical model. That is material mainly from
Section 7.1 in the book.
Lecture 28: M Dec 10, 2007
Quadratic forms and their associated symmetric matrix. Positive definite and positive semidefinite quadratic forms;
criteria for testing these properties. This is material from Section 8.2 in the book.
Lecture 29: W Dec 12, 2007
EXAM 3
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Last modified on Tuesday, 11-Dec-2007 21:24:51 CST