Math E-21b – Spring 2010 – Calendar of
topics and HW assignments
Last updated 14 Mar 2010 02:20 AM .
This Calendar will change as the course proceeds.
| Date | Topics | Text sections and homework assignments [Solutions (requires username/password)] | ||||||||
| Thurs, Jan 28 (Class #1) |
Algebra and geometry of lines, planes; solving simultaneously m equations in n unknowns; row reduction and row operations; reduced row echelon form; rank of a matrix; product of a matrix and a vector. |
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| Thurs, Feb 4 (Class #2) |
Vector and matrix forms of
systems of linear equations; linear transformations from Rm
to Rn defined by matrices; geometric meaning of
linearity; domain and codomain; meaning of the columns of a matrix;
rotations and dilations; shears; projections and
reflections. Inverse of a linear transformation and method for
finding the inverse of a (square) matrix.
Supplement
on the dot product and orthogonal projection |
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| Thurs, Feb 11 (Class #3) |
Inverse of a matrix; matrix
algebra; associativity and composition of linear functions; image and
kernel of a linear transformation; linear combinations and the span of a
set of vectors; subspaces; linear independence; basis of a subspace. Note: You may want to practice entering matrices on a calculator and performing matrix algebra with the calculator. |
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| Thurs, Feb 18 (Class #4) |
Test for linear independence, basis and dimension of a subspace, proof that dimension is well-defined; bases for kernels and images; Rank-Nullity Theorem; coordinates of a vector relative to a basis; matrix of a linear transformation relative to an alternate basis. |
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| Thurs, Feb 25 (Class #5) |
Matrix of a linear transformation relative to an alternate basis, applications to linear transformations defined geometrically. Intro. to general linear spaces, examples - continuous functions, differentiable functions, polynomials of degree less than or equal to n, the linear space of m by n matrices; complex numbers as a (real) 2-dimensional linear space. Subspaces of a linear space; basis and dimension; coordinates relative to a basis. General linear transformations and their matrices relative to a basis or bases. |
Note: Some of the details in Chapter 4 will be left to the reading. What you should get out of this chapter and the exercises is the sense that most constructions in Rn have analogues in the context of more general linear spaces and transformations. |
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| Thurs, Mar 4 (Class #6) |
Orthogonality
(perpendicularity) of vectors in Rn; length (norm) of
a vector, unit vectors; Cauchy-Schwartz inequality; orthogonal
complements and method for finding them; introduction to orthogonal
projections (see text for additional details).
Midterm Exam
#1 covering Chapters 1-3 of the text. |
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| Thurs, Mar 11 (Class #7) |
Orthogonal projections;
orthonormal basis; angle between two vectors; Gram-Schmidt
orthogonalization process; QR factorization. Orthogonal transformations
and orthogonal matrices. Least-squares approximation, normal equation;
data-fitting. Supplement on Least Squares in Economics |
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| Thurs, Mar 18 |
No Class - Harvard Spring Break |
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| Thurs, Mar 25 (Class #8) |
Determinant of a (square) matrix, patterns and permutations; Laplace expansion; multilinearity and the effect of the row operations on the value of the determinant; determinant criterion for invertibility of a matrix; k-volumes; determinant as an expansion factor; Cramer's Rule and formula for finding A-1 (minors, cofactors, and the classical adjoint). |
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| Thurs, Apr 1 (Class #9) |
Summary of facts about determinants and applications. Discrete (linear) dynamical systems, iteration of a matrix, trajectories and phase portraits; eigenvectors and eigenvalues of a (square) matrix; characteristic polynomial; algebraic and geometric multiplicities; diagonalization and the existence of a basis of eigenvectors; powers of a matrix. |
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| Thurs, Apr 8 (Class #10) |
Diagonalization and the existence of a basis of eigenvectors; powers of a matrix; trace and determinant; repeated eigenvalues; complex eigenvalues; review of facts about complex numbers; rotation-dilation matrices. |
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| Thurs, Apr 15 (Class #11) |
Rotation-dilation matrices and complex eigenvalues; eigenvalues and stability of a discrete linear dynamical system (phase portraits). Examples of complex and repeated eigenvalue cases. Spectral Theorem; symmetric matrices and diagonalization by an orthonormal basis. |
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| Thurs, Apr 22 (Class #12) |
Spectral Theorem; symmetric
matrices and diagonalization by an orthonormal basis. Quadratic forms;
positive definiteness of a matrix; principal axes; applications to
ellipses and hyperbolas.
Midterm Exam 2 covering chapters 4, 5.1 to 5.4, 6, and 7. |
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| Here's a website that has a good java-based tool for doing phase-plane analysis: http://math.rice.edu/~dfield/dfpp.html. Contrary to what it says on this page, you do not need MATLAB or any other software to use this tool. Choose the PPLANE option. You can enter new functions and change the size of the window. To see trajectories, just click on a point in the phase-plane. You should be able to print the phase portraits produced by this tool. | ||||||||||
| Thurs, Apr 29 (Class #13) |
Systems of linear differential
equations and their solutions - distinct real eigenvalue case, complex
eigenvalue case.
Supplement on 1st order linear systems of differential equations and the use of evolution matrices |
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| Thurs, May 6 (Class #14) |
Systems of linear differential equations and their solutions - complex eigenvalue examples, repeated eigenvalue case, and The Big Picture. Brief introduction to nonlinear systems and linearization around equilibria. |
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| Thurs, May 13 |
FINAL EXAM (location to be determined) |
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