Numerical Linear Algebra

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Chapter 7 Numerical Linear Algebra

Linear Algebra has two broad, but related concepts: solving linear systems and operations with matrices. This chapter goes into some detail about both of these from the point of view of numerical algorithms.

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Recall that in Subsection 2.4.5, there was an example of solving a 3 by 3 linear system by row-reducing a matrix, as is typically done ``by hand’’ in a Linear Algebra class. The example showed that the answer was quite susceptible to round-off errors. In Section 7.1, we will investigate ways to improve the techniques to solving these systems.

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Another category that is helpful in performing numerical linear algebra is that of performing a matrix factorization. One such technique is that of an $LU$-factorization in which the original matrix $A$ can be written as $A=LU\text{,}$ where $L$ is a lower-triangular matrix and $U$ is an upper triangular matrix.

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