making transition to linear independence through set theory and web module
Download
Skip this Video
Download Presentation
Making Transition to Linear Independence Through Set Theory and Web Module.

Loading in 2 Seconds...

play fullscreen
1 / 32

Making Transition to Linear Independence Through Set Theory and Web Module. - PowerPoint PPT Presentation


  • 74 Views
  • Uploaded on

Making Transition to Linear Independence Through Set Theory and Web Module. Hamide Dogan-Dunlap UTEP [email protected] ACTIVITY.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Making Transition to Linear Independence Through Set Theory and Web Module.' - eric-hensley


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
making transition to linear independence through set theory and web module

Making Transition to Linear Independence Through Set Theory and Web Module.

Hamide Dogan-Dunlap

UTEP

[email protected]

JMM January 2008, San Diego

activity
ACTIVITY

Goal is to have students conjecture on the characteristics of spanning sets for a given vector space, and gain an initial understanding of some aspects of the linear independence concept.

  • HMW ASSIGNMENT
    • Guided Questions
    • Experimental
  • WEB MODULE
    • Interactive
    • provides vector spaces In R3
    • Supported by
      • Mathematica
      • WebMathematica
        • This allows access to the module from any computer with Java application available.
    • Run on Java Platform
hmw assignment
HMW Assignment
  • Homework assignment has guided questions.
  • After experimenting on questions provided, students are asked to discuss various aspects of span and spanning sets.
  • Hmw questions are given using the set theoretical descriptions of vector spaces.
  • Click the next two slides to find some of the hmw questions.
web module
WEB MODULE
  • The module’s website is divided into two sections: instruction and experimental sections.
  • Instruction section provides information on how to enter vectors and the meaning of some of the symbols used.
  • The experimental section is the one that provides the dynamic geometrical outcome of vectors and vector spaces.
  • Before the hmw assignment, students are introduced to the module and geometrical interpretations of vector sum and scalar multiplications. With the particular activity, students are able to observe not only integer sums and multiples, but also rational number multiples of vectors.
slide7
This addresses the problem reported in literature. Many linear algebra students seem to consider sums and scalar multiples only for integer scalars, and not for rational number values.
  • See the next two slides for a demo on sums and scalar multiples of vectors.
  • On the second slide, one can observe that the red vector is about the half of the dark blue vector. In fact, students can verify that red is exactly half of the dark blue by entering a new vector (half of the dark blue) and comparing it with the red vector.
slide10
Before hmw questions are assigned, students go through various examples on how to determine the position of a vector using the vectors of a spanning set of a vector space.
  • For instance, as seen on the next slide, students are able to look at the geometrical representations of three vectors and their positions with respect to the plane spanned by the first two vectors, and determine that the red vector is the linear combination of the dark and light blue vectors. The linear combination furthermore can be read as red=-light blue+dark blue.
slide12
The module allows one to focus in and out of a graph.
  • This aspect makes it easier to study the various characteristics of the geometric representations of vectors and vector spaces.
  • Animate the next five slides to observe this aspect of the module.
slide18
The module also allows one to rotate the figure 360 degree in all directions. This makes it possible one to study the figures from multiple perspectives.
  • Animate the next six slides to observe this aspects of the module for four vectors and a plane spanned by the first two vectors.
slide25
Furthermore, the module can be modified to obtain multiple vector spaces in one figure. This allows one to investigate various geometrical aspects of three or more vectors in the presence of multiple vector spaces.
  • Animate the next two slides to observe this for two planes spanned by four vectors. The first two vectors are spanning one plane and the next two are spanning the other.
slide28
Also, observe that the module allows one to adjust the number of grids to be provided. For multiple figures, if one uses a higher number for grids then this may lead to a crowded outcome which may make it difficult to distinguish objects on a graph.
  • Click on the next two slides to observe this aspect of the module. The first slide shows the graph of four vectors and two vectors spaces with grid number 4, and the second slide shows the same graph with grid number 2.
slide31
This activity is followed by another activity on linear independence. The activity consists of two components:
    • Homework
      • Guided questions
      • Experimental
    • The same web module
slide32
Link to the Web-module can be found at

http://www.math.utep.edu/Faculty/hdogan/home.htm

ad