What are some applications of matrices, basic trigonometry, and linear systems in the real world? I'm writing and Algebra report about where these topics are used outside of the classroom.

Matrices are CENTRAL to much computer graphics. The transformations of 3-space for different points of view, the transformations from 3-space to 2-space, all of these are done with matrices. In fact, these matrix multiplications are so important, some computer graphics terminals are hard wired for matrix multiplications - just to speed things up.

The leading software package for working with matrices and solving linear systems of equations is Matlab and is put out by Mathworks. If you go to the following URL, you can see the differnt areas in which Mathworks does consulting.

http://www.mathworks.com/consulting/

Many of the big corporations, such as GM, Honeywell, 3M, Boeing, Xerox, Bell Labs, the transportation industries, ..., as well as government agencies involving national defense and resource allocation hire Linear Algebraists on their research teams. Almost anything can be modeled linearly.

Look at the books "Linear Algebra and its Applications" by David Lay, and "Matrix Analysis and Applied Linear Algebra" by Carl Meyer for specific examples and ideas that you can reference in your report.