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

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

Hamide Dogan-Dunlap

UTEP

JMM January 2008, San Diego

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

- 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

- 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.

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.

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.

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.

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.

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.

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.

This activity is followed by another activity on linear independence. The activity consists of two components:

- Homework
- Guided questions
- Experimental
- The same web module

Link to the Web-module can be found at

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

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