Linear Algebra

56 Chapter One. Linear SystemsThe underlying theme here is that one way to understand a mathematicalsituation is by being able to classify the cases that can happen. We have seenthis theme several times already. We classified solution sets of linear systemsinto the no-elements, one-element, and infinitely-many elements cases. We alsoclassified linear systems with the same number of equations as unknowns intothe nonsingular and singular cases. These classifications helped us understandthe situations that we were investigating. Here, where we are investigating rowequivalence, we know that the set of all matrices breaks into the row equivalenceclasses and we now have a way to put our finger on each of those classes — wecan think of the matrices in a class as derived by row operations from the uniquereduced echelon form matrix in that class.Put in more operational terms, uniqueness of reduced echelon form lets usanswer questions about the classes by translating them into questions about therepresentatives. For instance, we now (as promised in this section’s opening)can decide whether one matrix can be derived from another by row reduction.We apply the Gauss-Jordan procedure to both and see if they yield the samereduced echelon form.2.7 Example These matrices are not row equivalent( ) ( )1 −3 1 −3−2 6 −2 5because their reduced echelon forms are not equal.( ) ( )1 −3 1 00 0 0 12.8 Example Any nonsingular 3×3 matrix Gauss-Jordan reduces to this.⎛⎜1 0 0⎞⎟⎝0 1 0⎠0 0 12.9 Example We can describe all the classes by listing all possible reduced echelonform matrices. Any 2×2 matrix lies in one of these: the class of matrices rowequivalent to this, ( )0 00 0the infinitely many classes of matrices row equivalent to one of this type( )1 a0 0where a ∈ R (including a = 0), the class of matrices row equivalent to this,( )0 10 0