Chapter 1

A linear equation in n unknowns is an equation of the form:

$$ a_1x_1 + a_2x_2 + … + a_nx_n = b$$

where a_1, a_2, … a_n, and b are real numbers and x_1, x_2, … x_n are variables

A linear system of m equations in n unknowns is then a system of the form:

$$ a_11x_1 + a_12x_2 + … + a_1nx_n = b_1$$ $$ a_21x_1 + a_22x_2 + … + a_2nx_n = b_2$$ .

.

. $$ a_m1x_1 + a_m2x_2 + … + a_mnx_n = b_m$$

When there are real numbers that satisfy the system it is considered consistent and has a solution set. When there is not a set of numbers it is considered inconsistent and has no solution.

A system takes a triangular from if the kth equation the coefficients of the first k-1 variables area ll zero and the coefficient of x_k is nonzero (k = 1, … n)

$$ A = \begin{pmatrix} a_11 & a_12 & \ldots & a_1n \cr a_21 & a_22 & \ldots & a_2n \cr \vdots & \vdots & \ddots & \vdots \cr a_m1 & a_m2 & \ldots & a_mn \cr \end{pmatrix} $$

$$ A = \begin{pmatrix} {a_1}_1 & {a_1}_2 & \ldots & a_1n \cr a_21 & a_22 & \ldots & a_2n \cr \vdots & \vdots & \ddots & \vdots \cr a_m1 & a_m2 & \ldots & a_mn \cr \end{pmatrix} $$