TABLE OF CONTENTS Many problems encountered in applied mathematics are linear or can be approximated by linear systems which are, in general, computationally tractable. The corresponding mathematical subdiscipline is called linear algebra. At the heart of linear algebra are techniques, such as Gaussian elimination and the Gauss-Jordan algorithm, for computing solutions of linear systems. The main tool of linear algebra is matrices which help to arrange coefficients and describe operations.
Most finite linear systems can be described by matrices, a very useful short-hand notation which emphasizes the underlying linear structure and the interdependencies between the equations.
Literature: Atkinson, Boehm Prautzsch, Conte de Boor
A linear system is a set of equations of the form
where the a's are given real numbers and the x's are unknowns. The array A of the coefficients a i , k ,
is called an m n matrix. The matrix A contains the element a i , k in its ith row and kth column. Similarly, the a i can be written as an m 1 matrix or m column,
Consequently one has
where a k represents the kth column of A. Note that a scalar a can be viewed as a 1 1 matrix.
The n m matrix 
, defined by 
, is called the transpose
