Simultaneous Solution of Linear Equations

Linear equations have the form (m equations in n imknowns) 

The matrix A transforms the vector space X, defined by the vector X, into a subspace of the vector space B, defined by the vector b. In order to solve the equation Ax = b, we seek the solution vector x. Using the definition of the inverse of A, A A — J, we have 

If the rank A is equal to the number of unknowns, the solution is unique. If the rank is less than the number of unknowns, there are an infinite number of solutions (Carnahan et al., 1969). 

Although a large number of techniques exist for solving a well-defined set of linear equations (one having a unique solution), the most efficient methods are those based on the method of Gaussian elimination. 

In order to understand Gaussian elimination, consider the following set of three linear equations  

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