Linear Algebra

Section I. Solving Linear Systems 253.7 Lemma For a linear system, where ⃗p is any particular solution, the solutionset equals this set.{⃗p + ⃗h ∣ ∣ ⃗h satisfies the associated homogeneous system}Proof We will show mutual set inclusion, that any solution to the system is inthe above set and that anything in the set is a solution to the system. ∗For set inclusion the first way, that if a vector solves the system then it is inthe set described above, assume that ⃗s solves the system. Then ⃗s − ⃗p solves theassociated homogeneous system since for each equation index i,a i,1 (s 1 − p 1 ) + · · · + a i,n (s n − p n )= (a i,1 s 1 + · · · + a i,n s n ) − (a i,1 p 1 + · · · + a i,n p n ) = d i − d i = 0where p j and s j are the j-th components of ⃗p and ⃗s. Express ⃗s in the required⃗p + ⃗h form by writing ⃗s − ⃗p as ⃗h.For set inclusion the other way, take a vector of the form ⃗p + ⃗h, where ⃗psolves the system and ⃗h solves the associated homogeneous system and notethat ⃗p + ⃗h solves the given system: for any equation index i,a i,1 (p 1 + h 1 ) + · · · + a i,n (p n + h n )= (a i,1 p 1 + · · · + a i,n p n ) + (a i,1 h 1 + · · · + a i,n h n ) = d i + 0 = d iwhere p j and h j are the j-th components of ⃗p and ⃗h.QEDThe two lemmas above together establish Theorem 3.1. We remember thattheorem with the slogan, “General = Particular + Homogeneous”.3.8 Example This system illustrates Theorem 3.1.Gauss’s Methodx + 2y − z = 12x + 4y = 2y − 3z = 0−2ρ 1 +ρ 2−→x + 2y − z = 12z = 0y − 3z = 0shows that the general solution is a singleton set.⎛⎜1⎞⎟{ ⎝0⎠}0∗ More information on equality of sets is in the appendix.ρ 2 ↔ρ 3−→x + 2y − z = 1y − 3z = 02z = 0