Linear Algebra

48 Chapter One. Linear Systems1.4 Example⎛⎞2 6 1 2 5⎜⎟⎝0 3 1 4 10 3 1 2 5−→⎠ −ρ 2+ρ 3As a linear system this is(1/2)ρ 1 −(4/3)ρ 3 +ρ 2−→(1/3)ρ 2−(1/2)ρ 3so a solution set description is this.⎛⎜2 6 1 2 5⎞⎟⎝0 3 1 4 1⎠0 0 0 −2 4⎛−3ρ 2 +ρ 1 ⎜1 0 −1/2 0 −9/2⎞⎟−→ −→ ⎝0 1 1/3 0 3⎠−ρ 3 +ρ 10 0 0 1 −2x 1 − 1/2x 3 = −9/2x 2 + 1/3x 3 = 3x 4 = −2⎛ ⎞ ⎛ ⎞ ⎛ ⎞x 1 −9/2 1/2x 2S = { ⎜ ⎟⎝x 3 ⎠ = 3⎜ ⎟⎝ 0⎠ + −1/3⎜ ⎟⎝ 1⎠ x ∣3 x3 ∈ R}x 4 −2 0Thus echelon form isn’t some kind of one best form for systems. Other forms,such as reduced echelon form, have advantages and disadvantages. Instead ofpicturing linear systems (and the associated matrices) as things we operateon, always directed toward the goal of echelon form, we can think of them asinterrelated when we can get from one to another by row operations. The restof this subsection develops this relationship.1.5 Lemma Elementary row operations are reversible.Proof For any matrix A, the effect of swapping rows is reversed by swappingthem back, multiplying a row by a nonzero k is undone by multiplying by 1/k,and adding a multiple of row i to row j (with i ≠ j) is undone by subtractingthe same multiple of row i from row j.A ρ i↔ρ j ρ j ↔ρ i−→ −→ Akρ i (1/k)ρ ikρ i +ρ j −kρ i +ρ jA −→ −→ A A −→ −→ A(We need the i ≠ j condition; see Exercise 16.)QEDAgain, the point of view that we are developing, buttressed now by thislemma, is that the term ‘reduces to’ is misleading: where A −→ B, we shouldn’tthink of B as “after” A or “simpler than” A. Instead we should think of themas inter-reducible or interrelated. Below is a picture of the idea. It shows thematrices from the start of this section and their reduced echelon form version ina cluster as inter-reducible.