Lecture 16: Some remarks on pivot columns of a matrix Definition: Let A be an (n m) matrix with columns A = [v1, . . . , vm]. A Summary: Lecture 16: Some remarks on pivot columns of a matrix Definition: Let A be an (n × m) matrix with columns A = [v1, . . . , vm]. A column vk, 1 k m, is called a pivot column if there is a sequence of elementary matrices E1, . . . , EN such that Ae = EA = E1 · · · EN A is in echelon form and the k-th column of Ae contains a pivot element. The following practical question arises with this definition: There are many dif- ferent echelon forms for a matrix. How do we decide whether some column is a pivot column ? Assume Ae = EA is an echelon form for A. If the k-th column of Ae contains a pivot element then vk is a pivot column. But what if this is not the case ? There may be another echelon form Ae = E A where the k-th column has a pivot element. But how do we find the 'right' echelon form ? The answer is that we do not need to. Any echelon form does the job. Proposition: Let A = (aij) and B = (bij) be matrices in echelon form such that B can be transformed into A by row reduction, i.e. there are elementary matrices E1, . . . , EN such that A = E1 · · · EN B. If (ai1, . . . , aim) and (bi1, . . . , bim) are two nonzero rows then they have their pivot elements in the same column, i.e. ji = li where Collections: Mathematics