# Applets in Linear Algebra (Mathematics Resource Center)

Topic(s):

This is a large collection of Java applets for many topics in linear algebra, from basic vectors to systems of linear equations to vector spaces and linear transformations (and more!)

Identifier:
http://www.mathresource.iitb.ac.in/linear%20algebra/appletsla.html
Subject:
Rating:
Creator(s):
Prof. Inder K. Rana, Mathematics Resource Center, Dept Of Math, Indian Institute Of Technology, Bombay, India
Cataloger:
Publisher:
Indian Institute Of Technology website
Rights:
Prof. Inder K. Rana, Mathematics Resource Center, Dept Of Math, Indian Institute Of Technology, Bombay, India
Format Other:
Java may need to be enabled in the Java Console

### While many work well, some have typos and others do not work

Many of the applets are well done, although some contain typos and the wording is not always clear. Some applets don't work

### A wide array of applets of decent but not particularly exciting

A wide array of applets of decent but not particularly exciting quality.

### Chapter 3: Some links are not correct

Chapter 3 of the Mathematics Resource Center: A link is wrong. Clicking on Applet 3.2 takes you to Applet 3.3.

### Chapter 4: Poorly implemented and documented applets

Chapter 4 of the Mathematics Resource Center: Poorly implemented and documented applets to illustrate the linearity of a 2x2 determinant with respect to a row and to visualize the parallelepiped whose volume (up to sign) the determinant calculates. The directions are unclear, as is the required order of operations. Reloading may cause a phantom shaded triangle to appear in the window, only to disappear when you click on a rounded rectangle button to initiate an operation. The graphics look nice, but I think they're of minimal utility in a linear algebra course.

### Chapter 5: poor coordination; non-standard notation

Chapter 5 of the Mathematics Resource Center: While the three applets have been designed by the same person, they have almost no commonalities in terms of appearance, interaction, or functionality. They are essentially independent applets. The webpage for each applet contains a brief overview of the mathematics and instructions explaining how to use the applet. In addition to using non-standard and inconsistent notation, he mathematical typesetting is poor and difficult to read. The instructions are direct and clear, but it's not clear exactly how you would make a specific point. The independence applet is the only one that attempts to work in $R^3$,- and it is not successful. Note that the MRC page for linear algebra contains some bad links; while clicking on the text description took me to a different applet, clicking on the image took me to the correct applet.

### Chapter 6: display different aspects of linear transformations

Chapter 6 of the Mathematics Resource Center: The five applets ilustrate different aspects of linear transformations. The first tries to provide a geometric understand on what makes a transformation linear. All transformations are builtin, with only 1 nonlinear example. The explanation why this transformation is nonlinear is good, but the corresponding explanations why the others are linear are missing. Another applet shows how different linear transformations affect a set region. The only real interactivity is a slider for stretching, shears, and rotations. It would have been nice if these examples could emphasize the points made in the first applet. The applet for a matrix of a linear transformation has a nice description. The interaction is limited to choosing one of two bases for the domain and range spacesand 1 of 3 transformations; the resulting graphs are not as clear as they could be. For the rotations, the matrix appears in terms of sin(x) and cos(x) and there is no mention of x elsewhere in this applet. The graphics in the composite of a linear transformation applet are pretty good - but the accompanying description is sparse. It would be nice if more linear transformations could have been included. The final applet, about how a change of basis affects the matrix of a linear transformation works only on one specfic transformation and one specific change of basis. It appears to me that these applets have been designed primarily for use by the author, and not by a wider audience.

### Chapter 7: emphasize inner product and area of parallelogram

Chapter 7 of the Mathematics Resource Center: The four Java applets are based on the connection between the inner product and the area of a parallelogram. In particular, they provide visual representations for the definition of the inner product, its commutative and disstributive properties, and the Cauchy-Schwartz inequality. This consistency and uniformity is a marked improvement over previous chapters in the MRC that I have reviewed. The mathematical background is a little obscure; it would be better to simply refer to different regions by color instead of worrying about the names of the vertices (which are sometimesm not visible due to shading, etc.). Also, while visually appealing, I'm not sure that some of the animations serve any mathematical purpose. All four applets benefit from the fact that the user can select the vectors at will within the applet; the only trouble is that it can be difficult to make two vectors exactly parallel or orthogonal.

### Chapter 8: nicer that the ones I've reviewed in earlier chapters

Chapter 8 of the Mathematics Resource Center: The two Java applets provide two different takes on the process of creating an orthonormal basis from a collection of vectors. The first, "Constructing Orthonormal Basis", shows the algebraic steps to find an orthonormal basis for a fixed collection of 4 vectors in $\mathbf{R}^4$. In this example two of the vectors are parallel, so the basis contains only 3 vectors. There is no interactivity in this applet, just stepping through fixed steps. The second applet, "Gram Schmidt Ortho Normalization", starts with the general steps (algebraic and geometric) for 2 vectors in $\mathbf{R}^2$. While there are no instructions, I figured out that clicking on the graph defines the two vectors used in computations. The computations are nicely complemented by a graph of each computed vector. This applet is much nicer that the ones I've reviewed in earlier chapters of the MRC.

### Chapter 9: I don't see very many uses for these applets

Chapter 9 of the Mathematics Resource Center: The three Java applets explore i) isometries in $\mathbf{R}^2$, ii) isometries in $\mathbf{R}^3$, and iii) eigenvalues and eigenvectors. (The applet uses "isometrics" where they mean "isometries".) The first applet allows the user to create a polygon in $\mathbf{R}^2$ and an isometry and shows the image of the polygon under the isometry. It would be more effective to also see the image under a non-isometric transformation.The other two applets are really quite disappointing. I don't really know what they are expecting me to see in the $\mathbf{R}^3$ visualizer. The eigenvalue/vector applet is based on finding the vectors that map into a parallel vector under a linear transformation. The instructions are not completely clear, the graphical display appears to be partially overwritten with an empty box. To further limit the applicability of this applet, it works only with 7 fixed examples,. Even though the applet appears to be setup for the user to enter some information about the eigenvalues/vectors, I can't really see what is being checked; the "Explanations" leave a lot to be desired. I don't see very many uses for these applets.