# Interactive Online Modules for Matrix Algebra and Linear Algebra

## Linear Transformations Modules

#### Linear Transformations in R^{3} (WebMathematica)

Linear Transformations in R^{2} (Java)

Linear Transformations in R^{2} (WebMathematica)

Reflections (Java)

#### Sample Investigations:

Null space and range of linear transformations

#### Other Topics that Can Be Covered Using this Module:

- Eigenvalues
- Eigenvectors
- Null Space
- Range of a Linear Transformation
- Special Transformations
- Rotation Transformation
- Reflection Transformation
- Determinant
- Characteristic Polynomial
- Singular Matrix
- Non-Singular Matrix
- Diagonal Matrix

#### Module Description

The module is designed to teach the connection between a 2x2 matrix and a linear transformation of the plane.

A yellow unit square ABCD is displayed in the Cartesian plane. The matrix displayed in the left upper corner is the transformation matrix defined by identifying the images of the two unit vectors AD and AB. The user changes the matrix by dragging the points D' and B' to any place in the plane. The resulting pink parallelogram AB'C'D' is the image of the yellow square under the defined transformation.

Once the matrix is defined, the image of any vector can be obtained. This can be observed in the vector AE' which is the image of the vector AE. As the point E is dragged along the plane, the image point E' changes its position accordingly. The square and the parallelogram can be turned off by the "Hide Figures" button; and can be turned back on by the "Show Figures" button.

If the feature "Turn Traces On" is selected, one can create an arbitrary picture on the plane by moving the point E, and watch its image made by the point E'. The traces can be turned off by the "Turn Traces Off" button, and can be erased by clicking on a red "X" button in the right bottom corner of the window.

Additional displays on this module are: the determinant of the matrix, the ratio of the distance AE'/AE and the coordinates of the point E.

The "Reset" button sets back the matrix to the identity matrix.

The "Show/Hide Grid" buttons show/hide the integer grid on the plane.