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Section I. Solving Linear Systems 25Proof. We will show mutual set inclusion, that any solution to the system isin the 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 between 1 andn,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 iwhere p j and s j are the j-th components of ⃗p and ⃗s. We can write ⃗s − ⃗p as ⃗ h,where ⃗ h solves the associated homogeneous system, to express ⃗s in the required⃗p + ⃗ h form.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 it solves the given system: for any equation index i,= 0a 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 h j is the j-th component of ⃗ h.QEDThe two lemmas above together establish Theorem 3.1. We remember thattheorem with the slogan “General = Particular + Homogeneous”.3.9 Example This system illustrates Theorem 3.1.Gauss’ methodx + 2y − z = 12x + 4y = 2y − 3z = 0x + 2y − z = 1−2ρ 1 +ρ 2−→ 2z = 0y − 3z = 0shows that the general solution is a singleton set.⎛ ⎞{ ⎝ 1 0⎠}0∗ More information on equality of sets is in the appendix.x + 2y − z = 1ρ 2 ↔ρ 3−→ y − 3z = 02z = 0