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Gaussian Elimination and Vector Spaces

1. Discuss each of the three operations in Gaussian elimination. What do they have in common and how do they help solve systems of linear equations?
2. What is an Eigenvalue? What is an Eigenvector?


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Suppose you need to find numbers x, y, and z such that the following three equations are all simultaneously true:
2x + y &#8722; z = 8,
&#8722; 3x &#8722; y + 2z = &#8722; 11,
&#8722; 2x + y + 2z = &#8722; 3
This is called a system of linear equations for the unknowns x, y, and z. They are called linear because each term is either constant or is a constant times a single variable to the first power. The goal is to transform this system to an equivalent one so that we can easily read off the solution. The operations to transform a system of equations to another, whilst still preserving the solutions are as follows:
? multiply or divide a row by a non-zero number
? switch two rows
? add or subtract a (not necessarily integer) multiple of one row to another one
The strategy is as follows: eliminate x from all but the first equation, eliminate y from all but the second equation, and then ...

Solution Summary

The solution provides detailed theory for the problem in a 3-page Word document.