Microworld: Three
Views of Linear Equations
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Author:
Dan
Kalman
This is a web presentation of three interpretations of a linear system: as simultaneous scalar equations, as a single vector equation, and as a matrix-vector equation. For each view there is an interactive visualization exercise, embedded in a webpage which explains the context for the exercise. The first exercise is illustrated below.
In this exercise, two equations in two unknowns are defined by user input (typed into the colored boxes). A button click with the mouse produces the graphs of the equations. The object is to verify that a point in the plane satisfies one of the equations just when that point lies on the corresponding line. Of course, the solution to the pair of equations is the intersection of the lines. The user selects test points by clicking the graph with the mouse, and the results of testing each point in the equations is shown numerically in the yellow text window.
The second exercise looks like this:
This time the system is cast into a vector form. Here, the unknowns are viewed as scalar coefficients for two specific constant vectors, and the goal is to choose these coefficients so as to produce a vector result equal to a third given constant vector. The three constant vectors are entered by the user in the red, blue, and yellow boxes. A button click shows a graphical representation of the situation, with the three vectors depicted as red, blue, and yellow line segments. The user can stretch or shrink the red and blue vectors with the mouse, in effect selecting values for the unknown coefficients. The goal is to make the pale yellow resultant vector coincide with the bright yellow constant vector.
This is the third exercise:
Now the system is expressed as a matrix-vector equation. Here the matrix is made up of fixed numerical constants (entered by the user). The unknowns make up one vector, and a second vector is defined by specific numerical constants (also entered by the user). The object is to find a variable vector which, when multiplied by the matrix, produces the specified constant vector. On the graph, the variable vector appears as a red line, its product with the matrix is shown as a blue line, and the desired outcome is shown as a yellow line. The interaction involves dragging the red line with the mouse, while watching the blue line move in response. The goal is move the red line in such a way that the blue line exactly matches the yellow line.
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| - James E. White, Ph.D. , Library Director, | ||
| author of this website, Mathwright 2000, MindScapes, | ||
| MathwrightWeb, and Mathwright32 |
