Linear Algebra

Section III. Reduced Echelon Form 532.2 Lemma (Linear Combination Lemma) A linear combination of linearcombinations is a linear combination.Proof. Given the linear combinations c 1,1 x 1 + · · · + c 1,n x n through c m,1 x 1 +· · · + c m,n x n , consider a combination of thosed 1 (c 1,1 x 1 + · · · + c 1,n x n ) + · · · + d m (c m,1 x 1 + · · · + c m,n x n )where the d’s are scalars along with the c’s. Distributing those d’s and regroupinggives= (d 1 c 1,1 + · · · + d m c m,1 )x 1 + · · · + (d 1 c 1,n + · · · + d m c m,n )x nwhich is a linear combination of the x’s.QEDIn this subsection we will use the convention that, where a matrix is namedwith an upper case roman letter, the matching lower-case greek letter namesthe rows.⎛A =⎜⎝· · · α 1 · · ·· · · α 2 · · ·..· · · α m · · ·⎞⎟⎠⎛B =⎜⎝· · · β 1 · · ·· · · β 2 · · ·..· · · β m · · ·2.3 Corollary Where one matrix reduces to another, each row of the secondis a linear combination of the rows of the first.The proof below uses induction on the number of row operations used toreduce one matrix to the other. Before we proceed, here is an outline of the argument(readers unfamiliar with induction may want to compare this argumentwith the one used in the ‘General = Particular + Homogeneous’ proof). ∗ First,for the base step of the argument, we will verify that the proposition is truewhen reduction can be done in zero row operations. Second, for the inductivestep, we will argue that if being able to reduce the first matrix to the secondin some number t ≥ 0 of operations implies that each row of the second is alinear combination of the rows of the first, then being able to reduce the first tothe second in t + 1 operations implies the same thing. Together, this base stepand induction step prove this result because by the base step the propositionis true in the zero operations case, and by the inductive step the fact that it istrue in the zero operations case implies that it is true in the one operation case,and the inductive step applied again gives that it is therefore true in the twooperations case, etc.Proof. We proceed by induction on the minimum number of row operationsthat take a first matrix A to a second one B.∗ More information on mathematical induction is in the appendix.⎞⎟⎠