Getting students to interact with linear algebra content can be a daunting task, especially when you want them to recognize the algebraic and geometric meanings behind many of the concepts. We present a glimpse of the possible  here is a student voice to introduce what we mean by "the possible":
"I learned alot [sic] through the use of Eigenizer & Gridmaster & Transformer. Although they were difficult to use at first, because I did not know what I was looking for, and could not see the connections, by the end of the year, eigenizer taught me basically all of the connections with eigenvalues, eigenvectors, character poly & so forth. At first, I did not understand and had trouble following class discussions about eigenvalues & so forth, but after doing the project, everything came together because I could visually see the connection. I know this is a specific example, but I think all of the programs were helpful to see the linear algebra concepts in the program."
David E. Meel and Thomas A. Hern are in the Department of Mathematics and Statistics at Bowling Green State University.
In no way do we claim to provide a final word on the subject, but we provide
Some of the tools produced to study and visualize linear algebra concepts are described by
We have noticed in these sources a lack of standalone tools. For instance, the tools generated by the ATLAST project were designed as MATLAB applications, and others have been based on Mathematica, Maple, and other software systems. This usually requires the user to own or have access to a particular software product, which may not be available for all platforms. In addition, some basic skill in using the software is usually required.
We present in this paper several interactive webbased tools for visually exploring various linear algebra concepts, tools that are not constrained by the requirement of owning particular software. By building these tools with JavaSketchpad, the only requirement for a user is a Javaenabled browser. In particular, we present three tools, GridMaster, Transformer2D, and Eigenizer, and a couple of bonus items for advanced linear algebra.
Each figure below shows a small part of the JavaSketchpad window  click on each image or its title to see a picture of the full window.




Each of these tools permits students to interact with both algebraic and geometric representations of concepts such as change of bases, coordinate systems, linear transformations, eigenvalues, and eigenvectors. We include student reactions to enhance the discussion and identify the effects of using such tools as part of linear algebra explorations.
Getting students to interact with linear algebra content can be a daunting task, especially when you want them to recognize the algebraic and geometric meanings behind many of the concepts. We present a glimpse of the possible  here is a student voice to introduce what we mean by "the possible":
"I learned alot [sic] through the use of Eigenizer & Gridmaster & Transformer. Although they were difficult to use at first, because I did not know what I was looking for, and could not see the connections, by the end of the year, eigenizer taught me basically all of the connections with eigenvalues, eigenvectors, character poly & so forth. At first, I did not understand and had trouble following class discussions about eigenvalues & so forth, but after doing the project, everything came together because I could visually see the connection. I know this is a specific example, but I think all of the programs were helpful to see the linear algebra concepts in the program."
David E. Meel and Thomas A. Hern are in the Department of Mathematics and Statistics at Bowling Green State University.
In no way do we claim to provide a final word on the subject, but we provide
Some of the tools produced to study and visualize linear algebra concepts are described by
We have noticed in these sources a lack of standalone tools. For instance, the tools generated by the ATLAST project were designed as MATLAB applications, and others have been based on Mathematica, Maple, and other software systems. This usually requires the user to own or have access to a particular software product, which may not be available for all platforms. In addition, some basic skill in using the software is usually required.
We present in this paper several interactive webbased tools for visually exploring various linear algebra concepts, tools that are not constrained by the requirement of owning particular software. By building these tools with JavaSketchpad, the only requirement for a user is a Javaenabled browser. In particular, we present three tools, GridMaster, Transformer2D, and Eigenizer, and a couple of bonus items for advanced linear algebra.
Each figure below shows a small part of the JavaSketchpad window  click on each image or its title to see a picture of the full window.




Each of these tools permits students to interact with both algebraic and geometric representations of concepts such as change of bases, coordinate systems, linear transformations, eigenvalues, and eigenvectors. We include student reactions to enhance the discussion and identify the effects of using such tools as part of linear algebra explorations.
Our initial aim is to demonstrate several webbased tools designed to explore elementary linear algebra content. We know that readers may have different interests, and this webbased environment allows you to navigate in different ways. We have developed the following three different paths through the article:
You will be able to navigate to the next item on your selected path by page numbers stated at the bottom of each page. If you follow every page, you will be on the Researcher path. As you move between pages, you will stay in the same window, but each tool will open in a separate window, allowing you to explore while walking through the orientation, as well as attempt the sample activity.
Next or page: 4. GridMaster Tool and Sample Activity
Next page: 3. Discussion of GridMaster
In January of 2002, we discovered a paper entitled "Multigrid Graph Paper" (Bevis, 2002) that claimed multigrid paper (see Figure 1) would be helpful for students studying a variety of linear algebra concepts such as linear transformations, change of bases, and coordinate systems.
Figure 1. Multigrid paper 
In Figure 1 we can see that vector P has coordinates [1,3] with respect to the natural basis {e_{1}, e_{2}}, but with respect to {u_{1}, u_{2}}, the vector P has coordinates [1,1].
We saw the multigrid paper as a means of streamlining the Lay (2003) presentation of coordinate systems using two different graph papers. In particular, Lay (2003) described the use of two separate grids similar to those shown in Figures 2 and 3 when negotiating between two coordinate systems. For instance, the vector x has coordinates 1 and 6, typically expressed as with respect to the natural basis {e_{1}, e_{2}} (see Figure 2). In contrast, Figure 3 displays the vectors b_{1}, b_{2} and x, but in reference to a grid defined by b_{1} and b_{2}. The position of x has not moved, but .


Figure 2. Standard graph paper 
Figure 3. Bgraph paper 
After examining Bevis' article, which suggested overlaying multiple grids on a single sheet, we felt that static multigrid graph paper could be improved upon by using The Geometer's Sketchpad. The resultant webbased module, entitled GridMaster, permits students to model a portion of R^{2} and define vectors that coordinatize this vector space. Specifically, this first module was designed to help students recognize the meanings of different coordinate systems and change of bases, two topics identified as difficult for students to understand (Carlson, 1993).
In addition, we sought to address an observation of Hillel (2000) that some of the representations used in linear algebra can serve as obstacles to the development of student understanding. Hillel specifically identified that students have difficulty generalizing the notion of an ntuple as no longer representing a single vector but a potential representation of any other vector. This is of significant importance when students have to work across different bases. This observation motivated us to look for different ways, other than the purely computational MATLAB interactions we had used before, to cause students to interact with multiple representations simultaneously.
Next or or page: 4. GridMaster Tool and Sample Activity
You will want to open the GridMaster web page in order to understand the following discussion. (Note: It will take a while to download JavaSketchpad initially, so please be patient.)
Open GridMaster in new window
You can use GridMaster to construct electronic multigrid paper. Start by moving the green and blue vectors  click on a circle at the end of a vector and drag it to a new position. Then click on the appropriate button to display a grid, e.g., a green grid corresponding to the green vectors. You can resize the scale of the entire grid space by moving the point e(1) if you need more a larger or smaller range of values. In the upper left, you will see the coordinates of the point P (the red vector) with respect to the natural basis and then with respect to the bases [P]_{B} defined by the blue vectors and [P]_{G}defined by the green vectors.
Note: The "?" at the bottom righthand corner of the workspace is a link to Key Curriculum Press and its About JavaSketchpad web page.
Find the vector P determined by the given coordinate vector [P](u,v) and the given basis {u,v}.
Given the vector P and a basis {u,v}, find the coordinate vector [P](u,v).
Use the information provided to determine the missing element.
This last question begins to motivate a need to construct a changeofcoordinates matrix in order to determine more readily if a solution exists. Although there are an infinite number of solutions, determining a single one using GridMaster requires careful attention to the interaction of vectors and how they impact the movement of the grid lines and values of the coordinatized vectors.
Next or page: 6. Transformer2D Tool and Sample Activity
Next page: 5. Discussion of Transformer2D Tool
When students look at a matrix of transformation, at best they examine it as the coefficient matrix of a system of linear equations, and at worst they consider it a random array of numbers completely devoid of meaning. In addition, when we as teachers describe a matrix, we often read the elements rowbyrow, further separating the potential geometric considerations of column vectors. Consequently, students have a difficult time when faced with geometrically interpreting matrices of transformations other than simple ones involving positive and negative ones and zeros. In response, we began to search for a means to help students interact with matrices of transformations that would lead them to viewing the column vectors as having powerful information.
Our use of Sketchpad for investigation of linear transformations from R^{2} to R^{2} allows students to investigate the geometry underlying linear transformations as well as to grapple with the way vector spaces are mapped into other vector spaces. Getting students interested, exploring, and excited can lead them to making deep insights into the content and is one of the primary goals of these webbased tools. As students investigate a large number of examples in a short period of time, they are forced to revise their conjectures over and over again until settling upon insights that are compatible with mathematical definitions of the concepts.
Consequently, we designed the Transformer2D tool to get students to interact with the concept of linear transformation that acts as a "unifying and generalizing concept" permeating the whole of linear algebra (Dorier 1991, 1995; Dorier et al. 2000a, 2000b). A perspective on the geometric meaning of the column vectors in a matrix of transformation can potentially help students overcome some of their difficulties, as identified by Carlson (1993) and Meel (1999b), with related concepts such as null and column spaces.
We developed this tool over an extended period of time. During the Spring 2002 and Fall 2003 terms, we pilottested the webbased tools to determine students' reactions to them and to test the transparency of the interface. Based on findings from a beta version written in The Geometer's Sketchpad 3.0 (Jackiw, 1995), elements of the module were not sufficiently transparent. Students were frustrated that particular elements could be moved which should have been held static. This and other observations gave rise to our developing the webbased modules that could lock elements of the sketch. In addition, moving to webbased modules allowed students to access the modules remotely without having to own Geometer's Sketchpad. During pilot testing of Transformer2D we found that students needed a primer on how to interact with the interface and displayed difficulties with interpreting the information they were receiving from the module. For instance, one student stated:
"I found the programs very difficult to understand. I do not learn well by computers, but I do think they are a good way to visualize the problem. They helped me see what was going on. . . . I think in the future it would be helpful to hand out a ditto explaining the programs, so that students can become familiar with all the features the program has to offer."
Another student said:
"The most difficult/frustrating thing of working with the programs was not knowing what I was looking for, and making connections and discovering things on my own. However, this turned into the best thing at the end because it is a good feeling once you make the connection."
As a consequence, we present in this article the revised interface that is accompanied by a more thorough explanation of the role of particular buttons and features.
Next or or page: 6. Transformer2D Tool and Sample Activity
Using Transformer2D is considerably different from using GridMaster, in which all actions took place on a single graphical interface. In Transformer2D, the interface has been partitioned into different regions, each having its own purpose. The yellow box (upper left) controls the column vectors defining the matrix of transformation  the green vector is the first column and the blue vector is the second column. By grabbing and moving the ends of these two vectors, you can construct any 2x2 matrix. Clearly, varying the matrices' column vectors changes only the matrix. This encourages students to think about Ax as a linear combination of the columns of A and hopefully begin to recognize that the column space is the range of the transformation.
Below the yellow box is a box that controls the vector x. As you move x (the red vector) about the domain of the transformation, you can watch the image T(x) (the magenta vector) change in the large area to the right of the screen depicting the codomain. Partitioning the sketch in this way allows students to investigate separately the impact of changes to the matrix of transformation and of the value of the xvector. However, in order not to focus solely on the transformation of a single vector, we have provided a variety of other builtin graphical elements in the domain: a unit circle, a unit square, a unit grid, and an adjustable quadrilateral. (Change the shape of the quadrilateral by moving its vertices.) Students can view the transformations of each of these shapes by selecting the corresponding button in the domain sector. In addition, students can turn on or off a tracing feature that marks out the path of the transformed vector T(x) as x is varied. This feature allows students to carry out freehand investigations.
Open Transformer2D in new window
Warning : The Transformer2D page is best viewed in 1024 x 768 resolution or greater and may take up to a minute to load.
Notes:
This sample activity provides a guided exploration of a particular matrix of transformation and includes a set of questions that can be asked for any other matrix of transformation.
Given your exploration, answer the following questions:
In addition, Transformer2D can be used to find a matrix of transformation with particular properties such as:
Next or page: 7. Student Responses, Part 1
Each of these comments provides evidence of how students interact with the concept of linear transformations in manners atypical of classroom discussion. Even though some of the statements may reveal misinterpretations and misgeneralizations, the overarching goal of the tool is to help students experience the mathematics and then lead them to examine additional examples that help them recognize the misinterpretation or misgeneralization. For instance, the last comment came from a pair of students who at first examined a particular example and made a particular observation. Then, after changing the values of u and v, they noticed that the previous observation was incorrect and they adjusted their observation. Getting students to interact directly with the linear algebra concepts allows them to formulate conjectures and then test them.
For example, one student wrote in his email dialogue journal:
"I have been working on the problems presented in the transformer worksheet and had a breakthrough. I think I understand how the transformations effect [sic] the vectors. The unit grid portion of the program is very useful. I think I understand a little better how the signs in different areas change the grid and the different values of vectors affect the transformation."
Another student wrote,
"The project you had us do was very helpful. I learned a lot about transformations when you asked us to explain what was going on without knowing what a transformation was. Now if I were to go back and explain what was going on in each matrix, I would use terms like, reflection, contractions and expansions, shears (vertical and horizontal) and projections. When using the unit circle or unit square, you can see what each matrix does to it and therefore you can conclude some sort of transformation."
To get a better feel for how students interacted with the Transformer2D tool and its correspondent cognitivelyguided activity, we show on the next page a sample student response to the entire activity.
Next or page: 8. Student Responses, Part 2
Our purpose for the Transformer activity is to get students to explore, make conjectures, and adjust their conjectures. To give a sense of how students use Transformer2D, we provide the work of a randomly selected student named Jack.
The worksheet link on the preceding page downloads a onepage MS Word document with the full statement of the project. However, students were working with a multipage version with space to fill in their answers. We provide that version here as a PDF file.
Here are the parts of Jack's work as scanned images  each opens in a separate window, and you may want to have more than one open at a time.
Jack's described his reactions to the transformer project in his weekly journal entry:
"In class this week we started looking at transformation matrices, still in the form Ax = b. Now this is the same equation we have been using, but now we are looking at it in a different way. Going into the computer lab we started a project using the transformer on Dr. Meel's personal website. This was an interesting tool, because it allowed us to see the effects of different matrices on the range of values in R^2. The different kinds of transformations include reflections, contractions, expansions, vertical and horizontal shear, and projection onto the xaxis and yaxis.
"The project at first seemed a little overwhelming to be honest. I really had no clue how to tell what each matrix was doing by looking at them. But as I tried more and more matrices, and experimented to see what changes I made would create, it all started to make sense. After messing around for a while with the transformer and reading the book, I was able to go through the packet and do my best to answer the questions. Some of them I am not entirely sure I approached the correct way, but I am sure that I understand transformations a lot better in spite of that. So even if I am slightly off from the answers, I at least have an understanding of what transformations are and how to create some myself."
We are immersing students in a learning situation in which making sense of the environment is one major component as they grapple with their own limited perspectives and enhance their understandings of one or another concept piece by piece. In such an environment, they keep exploring, keep conjecturing, keep trying to organize thinking in new ways to accommodate the new bits of information being displayed, while at times experiencing considerable frustration because they lack the big picture. The tools and the corresponding projects are designed to eliminate the frustration and also to permit the teacher to assist, probe for understanding, point out significant hurdles, suggest alternative lines of thinking, and help equip students to manage their frustration and continue to pursue knowledge.
Next or page: 10. Eigenizer Tool and Sample Activity
Next page: 9. Discussion of Eigenizer Tool
Many students claim to have difficulty understanding the concepts of eigenvalues and eigenvectors (Meel, 1999b). One possible explanation for this difficulty could be that students typically associate eigenvalues and eigenvectors with a computational process and do not understand the geometric connections.
So, how do we encourage students to begin to look beyond the computations and consider the geometry? We consider an associated cognitivelyguided activity as the means of forcing students to come to grips with both the computational process and the underlying geometric relationship between eigenvalues and eigenvectors. Specifically, the activity integrates the capabilities of MATLAB (MATLAB, 1995) with the Eigenizer tool. By blending the 2D graphical examination of eigenvalues and eigenvectors through the Eigenizer tool with examination of higher dimensional eigenspaces through MATLAB, we lead students to explore eigenvalues and eigenvectors from multiple perspectives.
This experience stimulated one student to make the following comment:
I liked working with eigenizer and MATLAB simultaneously. That way, I could see what it would look like with l = 0 and l = a+bi. Also the eigenizer showed why an eigenvalue was (+) or () depending on which way the vectors pointed."
Another student stated
"I learned the most from the eigenizer project. The eigenizer helped you see that the formula was true, but you need to do the work in order to find all of the eigenvalues, not rely solely on the computer. Eigenizer needs to be able to do (3) vectors."
Looking at the Eigenizer tool and its cognitivelyguided activity as a whole, we see that students are being asked to look beyond the computations and understand the meanings behind the concepts eigenvalue, eigenvector, and eigenspace.
Next or or page: 10. Eigenizer Tool and Sample Activity
Working with Eigenizer, similar to Transformer2D, involves coordinated actions between defining the column vectors of the matrix of transformation. The yellow box controls the column vectors defining the matrix of transformation. In particular, the green vector controls the first column vector and the blue vector controls the second column vector. By grabbing the ends of these two vectors, you can construct any 2x2 matrix.
Below the yellow box is a box that controls the vector x. As you move x (the red vector) about the domain of the transformation, you can watch both x and the image T(x) (the magenta vector) change in the large area to the right of the screen, depicting the codomain of the transformation. The movement of the vector T(x) depends on the nature of the matrix of transformation.
The large codomain region also displays information concerning the length of the vector x, the length of the vector T(x), the radian measure of the angle between these two vectors, and a lambda approximator. Two buttons at the bottom of this region control the display of the Eigen Equations in a red box above the codomain box.
Open Eigenizer in new window
Note: The "?" at the bottom righthand corner of the workspace is a link to Key Curriculum Press and its About JavaSketchpad web page.
This sample activity provides a guided exploration of eigenvalues and eigenvectors of a particular matrix and includes a set of questions that can be asked for any other matrix, as well as some general questions about the tool and observations made from interacting with the tool.
Given your exploration (and perhaps some additional ones), answer the following questions:
The tool allows students to explore specific matrices as well as hypothesize about possible matrices with particular properties. It is with the latter type of explorations that the worlds of geometry and computation can fuse. Students need to think beyond computations that MatLab might be able to perform and ponder the possibilities of "what if?".
Next or page: 11. Student Responses
Clearly, some of the students were able to make observations that were generally consistent with mathematical definitions of eigenvalue and eigenvector and others made remarks that were slightly askew from such definitions.
However, being able to explore the geometry associated with eigenvectors and eigenvalues allowed some students to identify that MATLAB provided anomalous results when they encountered the following problem (2c in the accompanying activity):
Here are some sample solutions provided by students for the 2c matrix:
Our other samples are in a separate page because of their width.
Better understanding of both the computational and geometric roles of eigenvalues and eigenvectors, when combined with enhanced understandings of change of bases and linear transformations, sets the stage for investigations into diagonalization and potentially Singular Value Decompositions. One of our goals in having students explore the geometric perspective associated with eigenvalues and eigenvectors is to get students to begin to recognize that computations cannot be taken at face value but need to be examined within context for validity.
Next or page: 14. Transformer3D Tool and Sample Activity
Next page: 12. What is JavaSketchpad?
What we have done here is use an existing software product, The Geometer's Sketchpad (Jackiw, 2001), but in a way for which it was not originally designed. It is intended for geometric constructions, but we have used the geometric basis and the interactivity to build our tools. Moreover, the current version has an applet, JavaSketchpad, which allows most Sketchpad code to be run on a Javaenabled web browser. So we now have true platform and softwareindependence as well. The tools in this article download the JavaSketchpad applet and the accompanying code in a way that is transparent to the user. The only skill required is use of a web browser  a nearly universal skill these days.
According to Key Curriculum Press (2001, p. 231), "JavaSketchpad is a Sketchpad extension that allows you to place simple sketches inside web pages you publish on the Internet." This allows an Internet user to interact with the elements of the sketch by dragging points and pressing action buttons even if they do not have local access to Sketchpad. JavaSketchpad is a scaleddown version of Sketchpad and therefore not capable of all the constructions generally available through The Geometer's Sketchpad 4.01.
Two elements are necessary to build a JavaSketchpad in order to interact with the dynamic sketch. The first element is the HTML file that contains the descriptions necessary to render the web page, as well as the descriptions of the geometric construction and associated action buttons included in the sketch. The second element is the JavaSketchpad applet. This applet differs from that of a plugin which must be installed by the user prior to operating a particular application. The applet is held in the same directory as the HTML file and is downloaded at the same time the HTML file is accessed from the visitor's machine. In particular, "The applet  a separate set of files  provides the functionality that interprets this description (of the geometric construction), displays the figure in your visitor's browsers, and lets them interact with it" (Key Curriculum Press, 2001, p. 232). This simplifies the transfer and allows the user to interact seamlessly with the sketch without having to first download and install a plugin.
Next or page: 14. Transformer3D Tool and Sample Activity
Next page: 13. Overcoming JavaSketchpad's Limitations
Our construction of each tool required some additional work beyond the development via a Sketchpad 4.01 document. Due to JavaSketchpad's limited command structure, particular constructions available in Sketchpad were not replicable in JavaSketchpad. For instance, when attempting to use the coordinates of a point as part of a computation, JavaSketchpad would permit one to obtain the coordinates but would not separate the abscissa and ordinate. Consequently, we used trigonometric relationships and relating lengths to the unit length to compute the needed values.
Other structures unsupported by JavaSketchpad are arcs, angle bisectors, dashed lines, points on polygonal interiors, to name a few. In addition, the Geometer's Sketchpad to HTML converter does not provide seamless conversion but attempts to convert the elements of a sketch that are permissible. Generally, the converter identifies sequences of geometric constructions that are not supported either because the construction utilizes an illegal object construction or a parameter not understood by the converter. This in turn required us to reconsider the underlying construction and develop it in a way to avoid the conflict.
Perhaps one of the most difficult limitations of JavaSketchpad is its inherent 2D nature. Although it would be beneficial to have students interact with the geometry of linear algebra concepts in higher dimensions, creating appropriate tools to aid those investigations using Geometer's Sketchpad is problematic. We have attempted to overcome these limitations, and this continues to be a work in progress.
To understand the visual limitation of rendering 3D on a 2D presentation device, we developed a Transformer3D tool to try to address some of the conclusions of Sierpinska et al. (1999) and Sierpinska (2000), who indicated that students who interact with linear algebra concepts from solely a twodimensional perspective via Cabri geometry might misgeneralize concepts. Consequently, we thought appropriate and effective pedagogical activities are needed that integrate a variety of technological tools, such as webbased sketches with multiple dimensions, MATLAB activities, Maple or Mathematica investigations, and byhand computations, thereby allowing students to investigate relationships in many dimensions and to develop their personal and collective understandings.
Next or or page: 14. Transformer3D Tool and Sample Activity
Transformer3D was designed to complement Transformer2D by forcing students to grapple with transformations from R^{3} to R^{3}, but doing so required us to adjust the way the student controls the development of the column vectors of the matrix of transformation. In order to overcome the 2D nature of Sketchpad and JavaSketchpad, we developed a means of representing 3D space on a 2D platform. Consequently, a vector's elements are controlled by moving sliders along the various axes. These sliders can in turn be used to further illustrate descriptions of vectors with respect to the natural basis.
Even though Transformer3D is not as polished as Transformer2D, it did provide opportunities for students to grapple with what it meant to construct transformations from R^{3} to R^{3}. We designed Transformer3D to have a similar interface to that of Transformer2D. The controllers of the three column vectors of the matrix of transformation are maneuvered in the yellow box. The three vectors are controlled by sliders on the axes rather than by selecting the head of the vector and manipulating it in space. We implemented the sliders to allow for the three dimensions being rendered on a twodimensional plane.
Below the yellow box is an area depicting the domain of the matrix transformation. For ease, the domain has been restricted to just the positive portion of the three axes. We provide an "animate points" button in this area, which randomly moves the vector x around in the restricted domain.
Two areas have been color coded to help students orient themselves. The larger area to the right contains a depiction of the range of the matrix transformation. The three vectors, a1, a2, and a3, are shown, along with the transformation of x with respect to the transformation T. In addition, students can choose to show or hide a wire frame, as well as observe the transformation of the two colored areas from the domain.
Open Transformer3D in new window
Note: The "?" at the bottom righthand corner of the workspace is a link to Key Curriculum Press and its About JavaSketchpad web page.
Using Transformer3D, attempt to accomplish the following:
Note: Transformer3D is quite sensitive, and it is difficult to obtain exact values and construct matrices containing coordinate values of 0 for multiple entries in the same column vector. This is one of the major flaws of attempting to render 3D pictures in a 2D environment.
Next page: 17. WebSVD Tool and Sample Activity
Next or page: 15. Student Reactions
Having students observe how different components of the matrix A impacted the nature of the transformation was the goal of constructing Transformer3D. In addition, having the students be able to make conjectures and then examine them in multiple settings has the potential to be valuable.
Next page: 16. Discussion of Tools for Advanced Linear Algebra
One of the driving problems of advanced linear algebra, as encouraged by Strang (1988, p. 197), is the discussion of the singular value decomposition, for "the SVD has become fundamental in scientific computing." In particular, the SVD solves the problem of diagonalizing a matrix, which differs from an algebraoriented course that might use the Jordan canonical form to answer that question.
By considering the question
What is the shape of the image of the unit circle under the transformation T(x) = Ax ?
we can nudge students toward thinking about the geometry associated with the transformation, rather than relying entirely on computations.
For instance, if we consider the matrix
we see this answer:
These pictures provoke two questions:
In the early years, Tom Hern and the late Cliff Long used handdrawn pictures to motivate a discussion. Then, over time, technology allowed them to portray these pictures in a more professional manner  but still static in nature. Finally, they used MATLAB to draw the unit circle, and, as the unit circle was traced out, the image curve would be traced. Now, with the advent of JavaSketchpad , the static pictures have become dynamic, webbased tools, namely WebSVD and Hern & Long SVD, allowing students to explore the singular value decomposition and explain the components of the SVD.
Next or or page: 17. WebSVD Tool and Sample Activity
The WebSVD tool, which is similar to those previously discussed, was originally written in MATLAB as part of the ATLAST project (Leon, Herman, & Faulkenberry, 1996), but the MATLAB platform has been frustrating from crossplatform and availability points of view. The webbased tool development presented here has breathed new life into that effort.
Specifically, WebSVD contains components similar to those of Transformer2D, but, given its purpose, it has different elements. The yellow box still controls the column vectors defining the matrix of transformation. In particular, the green vector is the first column, and the blue vector is the second column. By grabbing the ends of these two vectors, you can construct any 2x2 matrix. Below the yellow box is a box that controls the vector x, restricted to the unit circle. As you move x (the red vector) about the unit circle in the domain of the transformation, you can observe the image T(x) (the magenta vector) change in the large area depicting the codomain of the transformation. In addition, the vector x can be moved automatically about the unit circle. In order to further explore, a variety of buttons are provided in the codomain portion containing T(x). For instance, you can turn on or turn off a tracing feature that marks out the path of the transformed vector T(x) or show/hide the orbit of T(x). Also, you can superimpose x onto the codomain, as well as create a separate vector p in the domain and its transformation T(p). Each of these elements helps support exploration into the singular value decomposition of a matrix of transformation.
Open WebSVD in new window
Note: The "?" at the bottom righthand corner of the workspace is a link to Key Curriculum Press and its About JavaSketchpad web page.
Consider the transformation T(x) = Ax with matrix
Using WebSVD, grab the red vector x (hold down mouse button) in the yellow box and move it completely around the unit circle. What shape do you seem to be getting?
To get a better look, click the "Turn Trace ON" button at the bottom of the right box, and move the red arrow around the unit circle. What shape so you seem to be getting? If you want, you can click on one of the "Animate x" buttons to have the vector x move automatically. If you want the animation to stop, click the button again.
To clean things up a little, click on the "Turn Trace OFF" button, and click the red X in the lower right hand corner of the screen. You can still view the image shape by clicking on "Show Orbit of T(x)" at the top of the right box. You can keep one of the "Animate x" buttons on if you wish (but not both).
Play for a while.
Experimentation should have led you to a feeling that the image of the unit circle appears elliptical. So, is it really an ellipse or not? If it is an ellipse, what are its axes?
A reasonable guess would be a set of eigenvectors from the matrix of transformation, A. Let's investigate that . . .
Now consider the following questions:
Note: Since this is a triangular matrix, its eigenvalues are the diagonal elements: l_{1} = 1 and l_{2} = 1.5. Corresponding unit eigenvectors are x_{1} = (1,0) and x_{2} = 1/sqrt(10) (3,1) = (.9487, .3162). These are shown in the following figure with Ax_{1} = x_{1} and Ax_{2} = 1.5 x_{2} in the image.
We've eliminated eigenvectors as the axes of the ellipse, perhaps there is another way to find the axes.
Once again, consider the transformation T(x) = Ax with matrix
If you need a hint, recall that x and p are eigenvectors of A^{T}A, and T(x) and T(p) are the corresponding eigenvectors of AA^{T}. The lengths of the vectors T(x) and T (p) are the square roots of the eigenvalues of A^{T}A. Note: You can check your answer by asking Matlab: Enter A, then type "[Q1,Sigma,Q2]=svd(A)".
Another means of examining singular value decompositions, described by Hern and Long (1991), is illustrated through the Hern & Long SVD tool, in which the decomposition is examined from the perspective of matrices of transformations that rotate, reflect, and stretch a unit circle to yield an ellipse. Rather than being an exploratory tool, this is an explanatory tool helpful for classroom discussions of the components of the SVD.
Consistent with the description of Hern and Long (1991), this tool contains a several regions that depict different aspects of the geometry associated with the singular value decomposition. As with most of our tools, the yellow box controls the column vectors defining the matrix of transformation. In particular, the green vector is the first column, and the blue vector is the second column. By grabbing the ends of these two vectors, you can construct any 2x2 matrix.
Next to the yellow box is the domain of the transformation that contains four vectors, x, p, v1, and v2. Vectors x and p are movable, with p restricted to the unit circle. Vectors v1 and v2 are restricted to the unit circle and dependent on the matrix of transformation. Specifically, v1 and v2 are the eigenvectors of the symmetric matrix A^{T}A. The transformation defined by the matrix V^{T} rotates the vectors v1 and v2 to the base vectors e_{1} and e_{2}. Then, the matrix S stretches these base vectors by the factors that are the lengths of the axes (and the positive square roots of the nonzero eigenvalues of AA^{T}). Finally, the matrix U rotates these to Av1 and Av2. Consequently, the matrix A can be written as V^{T}SU, where the entries s_{i} of S are called the singular values of A, and A = V^{T}SU is called the singular value decomposition of A.
Open Hern & Long SVD in new window
Note: The "?" at the bottom righthand corner of the workspace is a link to Key Curriculum Press and its About JavaSketchpad web page.
Next or or page: 19. Conclusion
Our goal in this paper is to share the possible. Developing webbased linear algebra tools has been a long process and one fraught with its ups and downs. The expertise necessary to develop Java applets is enormous, and the learning curve is steep. This has left many educators with ideas but without the means to express them. When Key Curriculum Press introduced JavaSketchpad , they provided educators with a means of publishing Java applets that was less complicated although more restricted than programming completely in the Java language.
Our various implementations of Transformer, GridMaster, Eigenizer, WebSVD, and Hern & Long SVD point to the potential efficacy of implementing webbased modules and correspondent cognitivelyguided activities as part of linear algebra instruction. As a consequence, we have now reached a point where we have a truly viable way to construct and implement widely available interactive tools. The tools must of course be accompanied by welldesigned activities that take advantage of both the power and limitations of the tools in order to help students learn mathematical content. Therefore, we see that the focus must now turn to the more difficult and lesserstudied topic: How to design materials and situations to take best advantage of such technology. Up until now this has been mostly anecdotal.
Next or or page: 20. References
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