## SIMULTANEOUS LINEAR EQUATIONS

Throughout this section, A will denote an n X n square matrix and X a matrix that is not necessarily square. Generally, we shall assume that X is n X К with К ^ n.

Consider the following n linear equations:

anxi + a12x2 + ■ ■ ■ + alnxn = y

CL%X} T CL22X2 T ‘ ‘ ‘ T Cl2nXn y2

(11.4.1)

ani*i "f "f ■ ■ ■ T annxn yn

Define x = (xj, x2,. . . , x„)’ and у = (уь Уъ • • • > Уп)’ an(l let A be as in

(11.1.1) with n = m. Then (11.4.1) can be written in matrix notation as

(11.4.1) Ax = y.

A major goal of this section is to obtain a necessary and sufficient condition on A such that (11.4.2) can be solved in terms of x for any y. Using the notation 3 (there exists), V (for any), and s. t. (such that), we can express the last clause of the previous sentence as

V у 3 x s. t...

Read More