Systems of Linear Equations, Matrices

Equations of the form \({\sum{a_i}{x_i}}\,=\,b, \) for unknowns x i with arbitrary given numbers a i and b, are called linear, and every set of simultaneous linear equations is called a linear system. They are generalizations of the equations of lines and planes which we have studied in Section 1.3. In this section, we begin to discuss how to solve them, that is, how to find numerical values for the x i that satisfy all the equations of a given system. We also examine whether a given system has any solutions and, if so, then how we can describe the set of all solutions.