Linear Algebra

2 Chapter One. Linear Systemsmust equal the number present afterward. Applying that in turn to the elementsC, H, N, and O gives this system.7x = 7z8x + 1y = 5z + 2w1y = 3z3y = 6z + 1wBoth examples come down to solving a system of equations. In each system,the equations involve only the first power of each variable. This chapter showshow to solve any such system.I.1 Gauss’s Method1.1 Definition A linear combination of x 1 , . . . , x n has the forma 1 x 1 + a 2 x 2 + a 3 x 3 + · · · + a n x nwhere the numbers a 1 , . . . , a n ∈ R are the combination’s coefficients. A linearequation in the variables x 1 , . . . , x n has the form a 1 x 1 + a 2 x 2 + a 3 x 3 + · · · +a n x n = d where d ∈ R is the constant.An n-tuple (s 1 , s 2 , . . . , s n ) ∈ R n is a solution of, or satisfies, that equationif substituting the numbers s 1 , . . . , s n for the variables gives a true statement:a 1 s 1 + a 2 s 2 + · · · + a n s n = d. A system of linear equationsa 1,1 x 1 + a 1,2 x 2 + · · · + a 1,n x n = d 1a 2,1 x 1 + a 2,2 x 2 + · · · + a 2,n x n = d 2.a m,1 x 1 + a m,2 x 2 + · · · + a m,n x n= d mhas the solution (s 1 , s 2 , . . . , s n ) if that n-tuple is a solution of all of the equationsin the system.1.2 Example The combination 3x 1 + 2x 2 of x 1 and x 2 is linear. The combination3x 2 1 + 2 sin(x 2) is not linear, nor is 3x 2 1 + 2x 2.1.3 Example The ordered pair (−1, 5) is a solution of this system.In contrast, (5, −1) is not a solution.3x 1 + 2x 2 = 7−x 1 + x 2 = 6Finding the set of all solutions is solving the system. We don’t need guessworkor good luck; there is an algorithm that always works. This algorithm is Gauss’sMethod (or Gaussian elimination or linear elimination).