A COLORFUL Linear Combination Demo

• Objective
• Level
• Prerequisites
• Platform
• Instructor's Notes
• Credits

•

Objective: Near the beginning of a linear algebra course the basic operations of matrix addition and scalar multiplication are combined to define linear combinations of vectors or matrices. We stress that linear combinations are the tools for building other objects.  We can show this geometrically in R2 and R3, but this colorful demo provides a realistic example of linear combinations in action.

Level:  A beginning linear algebra course.

Prerequisites: Students should be familiar with the basic operations of matrix addition and scalar multiplication and the notion of a linear combination.

Platform: A browser that supports Java or a  computer system with MATLAB .

The Java Applet which is in RGBExample.zip and the MATLAB m-file rgbexamp provide a demonstration of colors as a linear combination of RED, GREEN, and BLUE using their intensities as coefficients.  Sliders control the selection of the intensity of each color.  After moving the sliders to the desired intensity, in MATLAB click the apply button to see the color formed and a display of the linear combination. In the Java Applet the colors change automatically as the sliders move and are released. A sample screen from the MATLAB implementation is shown below.

In Netscape, press shift and simultaneously click on the file to save this file to  your disk.  Make sure you place the file in a directory that is in the MATLAB path.

(MATLAB v 5.3 Release 11)

To experiment with the Java Applet click on RGBJAVA

which must be unzipped in order to execute the routine.

Instructor's Notes:

A display screen S is really a matrix of pixels and each entry of matrix S has a 3-vector of RGB intensities associated with it.  Let R, G, and B be matrices of the same size as S, where matrix R contains the red intensities of the pixels of S, G the green intensities, and B  the blue intensities.  Thus the configuration of colors shown on screen S depends upon the information in the three matrices R, G, and B

Since the color of each pixel is a blend of the three light sources, this situation amounts to saying that

S = R + G + B.

At a single pixel, the color of the pixel is determined by a linear combination of the 3-vectors in

C = {[1 0 0], [0 1 0], [0 0 1]}

where the coefficients, the RGB intensities, are restricted to have values between 0 and 1. For example in MATLAB,

The accompanying Java Applet in RGBExample.zip and the MATLAB m-file, rgbexamp.m, let students or instructors experiment with linear combinations to form colors. A discussion on how to form, white, black, red, green, and blue by selecting intensities has been helpful in providing students with a useful example of linear combinations.

Credits:  This demo and the MATLAB m-file were submitted by

Dr. David R. Hill
Department of Mathematics
Temple University

and is included in Demos with Positive Impact with his permission. The Java Applet was written by Joshua Kraft, a student at Temple University, and is included with his permission.

DRH 5/7/01   Last updated 5/23/2006

Since 3/1/2002