Page 752 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Solution (3)

Rewrite 3 as (4)
(5)
As specified in Step 2 above, define , , and by the equation

so 4 can be rewritten as

or, equivalently,

The procedure for solving this system is similar to back-substitution except that the equations are solved from the top down
instead of from the bottom up. This procedure, which is called forward-substitution, yields
(verify). Substituting these values in 5 yields the linear system

or, equivalently,

Solving this system by back-substitution yields the solution

(verify).

-Decompositions

Now that we have seen how a linear system of n equations in n unknowns can be solved by factoring the coefficient matrix,
we shall turn to the problem of constructing such factorizations. To motivate the method, suppose that an matrix A has
been reduced to a row-echelon form U by Gaussian elimination—that is, by a certain sequence of elementary row operations.
By Theorem 1.5.1 each of these operations can be accomplished by multiplying on the left by an appropriate elementary
matrix. Thus there are elementary matrices , , …, such that

(6)