Linear Algebra

Section III. Reduced Echelon Form 55With B and C different only in column n, suppose that they differ in row i.Subtract row i of (∗∗∗) from row i of (∗∗) to get the equation (b i,n −c i,n )·x n = 0.We’ve assumed that b i,n ≠ c i,n so the system solution includes that x n = 0.Thus in (∗∗) and (∗∗∗) the n-th column contains a leading entry, or else thevariable x n would be free. That’s a contradiction because with B and C equal onthe first n − 1 columns, the leading entries in the n-th column would have to bein the same row, and with both matrices in reduced echelon form, both leadingentries would have to be 1, and would have to be the only nonzero entries inthat column. Thus B = C.QEDThat result answers the two questions that we posed in the introduction tothis section: do any two echelon form versions of a linear system have the samenumber of free variables, and if so are they exactly the same variables? We getfrom any echelon form version to the reduced echelon form by pivoting up, andso uniqueness of reduced echelon form implies that the same variables are freein all echelon form version of a system. Thus both questions are answered “yes.”There is no linear system and no combination of row operations such that, say,we could solve the system one way and get y and z free but solve it another wayand get y and w free.We end this section with a recap. In Gauss’s Method we start with a matrixand then derive a sequence of other matrices. We defined two matrices to berelated if we can derive one from the other. That relation is an equivalencerelation, called row equivalence, and so partitions the set of all matrices intorow equivalence classes.( 1 3)2 7( 1 3)0 1. . .(There are infinitely many matrices in the pictured class, but we’ve only gotroom to show two.) We have proved there is one and only one reduced echelonform matrix in each row equivalence class. So the reduced echelon form is acanonical form ∗ for row equivalence: the reduced echelon form matrices arerepresentatives of the classes.⋆⋆⋆ ( 1 0)0 1. . .⋆∗ More information on canonical representatives is in the appendix.