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Linear Algebra II
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Linear Algebra is both rich in theory and full of interesting applications; ...

Linear Algebra is both rich in theory and full of interesting applications; in this course the student will try to balance both. This course includes a review of topics learned in Linear Algebra I. Upon successful completion of this course, the student will be able to: Solve systems of linear equations; Define the abstract notions of vector space and inner product space; State examples of vector spaces; Diagonalize a matrix; Formulate what a system of linear equations is in terms of matrices; Give an example of a space that has the Archimedian property; Use the Euclidean algorithm to find the greatest common divisor; Understand polar form and geometric interpretation of the complex numbers; Explain what the fundamental theorem of algebra states; Determine when two matrices are row equivalent; State the Fredholm alternative; Identify matrices that are in row reduced echelon form; Find a LU factorization for a given matrix; Find a PLU factorization for a given matrix; Find a QR factorization for a given matrix; Use the simplex algorithm; Compute eigenvalues and eigenvectors; State Shur's Theorem; Define normal matrices; Explain the composition and the inversion of permutations; Define and compute the determinant; Explain when eigenvalues exist for a given operator; Normal form of a nilpotent operator; Understand the idea of Jordan blocks, Jordan matrices, and the Jordan form of a matrix; Define quadratic forms; State the second derivative test; Define eigenvectors and eigenvalues; Define a vector space and state its properties; State the notions of linear span, linear independence, and the basis of a vector space; Understand the ideas of linear independence, spanning set, basis, and dimension; Define a linear transformation; State the properties of linear transformations; Define the characteristic polynomial of a matrix; Define a Markov matrix; State what it means to have the property of being a stochastic matrix; Define a normed vector space; Apply the Cauchy Schwarz inequality; State the Riesz representation theorem; State what it means for a nxn matrix to be diagonalizable; Define Hermitian operators; Define a Hilbert space; Prove the Cayley Hamilton theorem; Define the adjoint of an operator; Define normal operators; State the spectral theorem; Understand how to find the singular-value decomposition of an operator; Define the notion of length for abstract vectors in abstract vector spaces; Define orthogonal vectors; Define orthogonal and orthonormal subsets of R^n; Use the Gram-Schmidt process; Find the eigenvalues and the eigenvectors of a given matrix numerically; Provide an explicit description of the Power Method. (Mathematics 212)

Subject:
Algebra
Material Type:
Full Course
Provider:
The Saylor Foundation
Linear Functions
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This task requires students to use the fact that on the graph ...

This task requires students to use the fact that on the graph of the linear function h(x)=ax+b, the y-coordinate increases by a when x increases by one. Specific values for a and b were left out intentionally to encourage students to use the above fact as opposed to computing the point of intersection, (p,q), and then computing respective function values to answer the question.

Subject:
Mathematics
Functions
Material Type:
Activity/Lab
Provider:
Illustrative Mathematics
Provider Set:
Illustrative Mathematics
Author:
Illustrative Mathematics
Linear Or Exponential?
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This task gives a variet of real-life contexts which could be modeled ...

This task gives a variet of real-life contexts which could be modeled by a linear or exponential function. The key distinguishing feature between the two is whether the change by equal factors over equal intervals (exponential functions), or by a constant increase per unit interval (linear functions).

Subject:
Mathematics
Functions
Material Type:
Activity/Lab
Provider:
Illustrative Mathematics
Provider Set:
Illustrative Mathematics
Author:
Illustrative Mathematics
Linear Pair of Angles
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An interactive applet and associated web page that demonstrate a linear pair ...

An interactive applet and associated web page that demonstrate a linear pair of angles. A pair of angles are shown and the user can drag the common side. It demonstrates that they a supplementary (add to 180 degrees) and have a common side. The angle measures can be turned off for class discussion. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

Subject:
Geometry
Material Type:
Reading
Simulation
Provider:
Math Open Reference
Author:
John Page
Linear Vector Spaces
Conditions of Use:
No Strings Attached
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Introduces tools and formulas to use when dealing with Linear Vector Spaces. ...

Introduces tools and formulas to use when dealing with Linear Vector Spaces. Topics covered include: linear vector spaces, inner product spaces, norm, Schwarz inequality, and distance between two vectors

Material Type:
Reading
Syllabus
Provider:
Rice University
Provider Set:
Connexions
Author:
Don Johnson
Linear function graph display
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An interactive applet that allows the user to graphically explore the properties ...

An interactive applet that allows the user to graphically explore the properties of a linear functions. Specifically, it is designed to foster an intuitive understanding of the effects of changing the two coefficients in the function y=ax+b. The applet shows a large graph of a quadratic (ax + b) and has two slider controls, one each for the coefficients a and b. As the sliders are moved, the graph is redrawn in real time illustrating the effects of these variations. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

Subject:
Geometry
Material Type:
Reading
Simulation
Provider:
Math Open Reference
Author:
John Page
Modeling Dynamics and Control I, Spring 2005
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First of two-term sequence on modeling, analysis and control of dynamic systems. ...

First of two-term sequence on modeling, analysis and control of dynamic systems. Mechanical translation, uniaxial rotation, electrical circuits and their coupling via levers, gears and electro-mechanical devices. Analytical and computational solution of linear differential equations and state-determined systems. Laplace transforms, transfer functions. Frequency response, Bode plots. Vibrations, modal analysis. Open- and closed-loop control, instability. Time-domain controller design, introduction to frequency-domain control design techniques. Case studies of engineering applications.

Subject:
Functions
Material Type:
Full Course
Textbook
Provider:
M.I.T.
Provider Set:
M.I.T. OpenCourseWare
Author:
Dubowsky, Steven
Trumper, David L.
Newton's Law of Cooling
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The coffee cooling experiment is a popular example of an exponential model ...

The coffee cooling experiment is a popular example of an exponential model with immediate appeal. The model is realistic and provides a good context for students to practice work with exponential equations.

Subject:
Mathematics
Functions
Material Type:
Activity/Lab
Provider:
Illustrative Mathematics
Provider Set:
Illustrative Mathematics
Author:
Illustrative Mathematics
PEMDAS: Algebraic Order of Operations
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Watch this music video to help you learn about PEMDAS (Please Excuse ...

Watch this music video to help you learn about PEMDAS (Please Excuse My Dear Aunt Sally). Does this sound familiar? If not, this is an excellent device to memorize the algebraic order of operations. This video is produced by Mr. Davis Productions and plays music by Odyssey Sound Lab.

Subject:
Algebra
Material Type:
Reading
Provider:
SMARTR
Provider Set:
SMARTR: Virtual Learning Experiences for Youth
Review Worksheets: Linear and Quadratice Equations, Unit Conversion, Significant Figures, and Trigonometry- Barbara Gilbert, CNM
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These worksheets, provided by Barbara Gilbert, review mathematics concepts that are necessary ...

These worksheets, provided by Barbara Gilbert, review mathematics concepts that are necessary in physics, including linear equations, quadratic equations, unit conversion, significant figures, and trigonometry.

Subject:
Applied Science
Mathematics
Material Type:
Homework/Assignment
Provider:
OER Commons
Provider Set:
Open Author Resources
Author:
Sonya Bennett-Brandt
Rising Gas Prices Đ Compounding and Inflation
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The purpose of this task is to give students an opportunity to ...

The purpose of this task is to give students an opportunity to explore various aspects of exponential models (e.g., distinguishing between constant absolute growth and constant relative growth, solving equations using logarithms, applying compound interest formulas) in the context of a real world problem with ties to developing financial literacy skills.

Subject:
Mathematics
Functions
Material Type:
Activity/Lab
Provider:
Illustrative Mathematics
Provider Set:
Illustrative Mathematics
Author:
Illustrative Mathematics
Running Races
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In this unit of five lessons from Illuminations, learners begin with a ...

In this unit of five lessons from Illuminations, learners begin with a number-line model and extend it to investigate linear relationships with the Distance, Speed, and Time Simulation from NCTM's E-Examples. Students then progress to plotting points and graphing linear functions while continually learning and reinforcing basic multiplication facts. Instructional plan, questions for the students, assessment options, extensions,and teacher reflections are given for each lesson as well as links to download all student resources.

Subject:
Mathematics
Algebra
Functions
Geometry
Material Type:
Interactive
Lesson Plan
Provider:
National Council of Teachers of Mathematics
Provider Set:
Illuminations
Author:
Grace M. Burton