This 11-minute video lesson proves the "associative," "distributive," and "commutative" properties for vector dot products.
This 23-minute video lesson looks at Rowspace and Left Nullspace. [Linear Algebra playlist: Lesson 97 of 143]
This 9-minute video lesson provides definitions of the vector dot product and vector length.
This 19-minute video lesson proves the triangle inequality for vectors in Rn.
This 20-minute video lesson looks at the relationship between left nullspace, rowspace, column space and nullspace. [Linear Algebra playlist: Lesson 98 of 143]
After being traditionally published for many years, this formidable text by W. Keith Nicholson is now being released as an open educational resource and part of Lyryx with Open Texts! Supporting today’s students and instructors requires much more than a textbook, which is why Dr. Nicholson opted to work with Lyryx Learning.
Overall, the aim of the text is to achieve a balance among computational skills, theory, and applications of linear algebra. It is a relatively advanced introduction to the ideas and techniques of linear algebra targeted for science and engineering students who need to understand not only how to use these methods but also gain insight into why they work.
The contents have enough flexibility to present a traditional introduction to the subject, or to allow for a more applied course. Chapters 1–4 contain a one-semester course for beginners whereas Chapters 5–9 contain a second semester course. The text is primarily about real linear algebra with complex numbers being mentioned when appropriate (reviewed in Appendix A).
Students use latex tubes and bicycle pumps to conduct experiments to gather data about the relationship between latex strength and air pressure. Then they use this data to extrapolate latex strength to the size of latex tubing that would be needed in modern passenger sedans to serve as hybrid vehicle accelerators, thus answering the engineering design challenge question posed in the first lesson of this unit. Students input data into Excel spreadsheets and generate best fit lines by the selection of two data points from their experimental research data. They discuss the y-intercept and slope as it pertains to the mathematical model they generated. Students use the slope of the line to interpret the data collected. Then they extrapolate with this information to predict the latex dimensions that would be required for a full-size hydraulic accumulator installed in a passenger vehicle.
This task asks students to consider the linear and quadratic functions appearing on a coordinate plane.
This 14-minute video lesson provides an Introduction to linear transformations.
Students measure the relative intensity of a magnetic field as a function of distance. They place a permanent magnet selected distances from a compass, measure the deflection, and use the gathered data to compute the relative magnetic field strength. Based on their findings, students create mathematical models and use the models to calculate the field strength at the edge of the magnet. They use the periodic table to predict magnetism. Finally, students create posters to communicate the details their findings. This activity guides students to think more deeply about magnetism and the modeling of fields while practicing data collection and analysis. An equations handout and two grading rubrics are provided.
Students apply high school-level differential calculus and physics to the design of two-dimensional roller coasters in which the friction force is considered, as explained in the associated lesson. In a challenge the mirrors real-world engineering, the designed roller coaster paths must be made from at least five differentiable functions that are put together such that the resulting piecewise curving path is differentiable at all points. Once designed mathematically, teams build and test small-sized prototype models of the exact designs using foam pipe wrap insulation as the roller coaster track channel with marbles as the ride carts.
Matrix Algebra with Computational Applications is a collection of Open Educational Resource (OER) materials designed to introduce students to the use of Linear Algebra to solve real-world problems. These materials were developed specifically for students and instructors working in a "flipped classroom" model that emphasizes hands-on problem-solving activities during class meetings, with students watching lectures and completing readings and assignments outside of the classroom. To access the Matrix Algebra with Computational Applications website, please go to http://colbrydi.github.io/MatrixAlgebra
Showing that matrix products are associative. Created by Sal Khan.
Students obtain a basic understanding of microfluidic devices, how they are developed and their uses in the medical field. After conducting the associated activity, they watch a video clip and learn about flow rate and how this relates to the speed at which medicine takes effect in the body. What they learn contributes to their ongoing objective to answer the challenge question presented in lesson 1 of this unit. They conclude by solving flow rate problems provided on a worksheet.
In this task students are asked to write two expressions from verbal descriptions and determine if they are equivalent. The expressions involve both percent and fractions. This task is most appropriate for a classroom discussion since the statement of the problem has some ambiguity.
The purpose of this task is to emphasize the use of the Remainder Theorem (a discussion of which should obviously be considered as a prerequisite for the task) as a method for determining structure in polynomial in equations, and in this particular instance, as a replacement for division of polynomials.
This task assumes students are familiar with mixing problems. This approach brings out different issues than simply asking students to solve a mixing problem, which they can often set up using patterns rather than thinking about the meaning of each part of the equations.
The problem deals with a rational expression which is built up from operations arising naturally in a context: adding the volumes of the fertilizer and the water, and dividing the volume of the fertilizer by the resulting sum. Thus it encourages students to see the expression as having meaning in terms of numbers and operations, rather than as an abstract arrangement of symbols.
This undergraduate course focuses on traditional algebra topics that have found greatest application in science and engineering as well as in mathematics.
Students will breed fruit flies through several generations and record their data using mathematical models in order to demonstrate the inheritance of trait variations.