Linear Algebra

52 Chapter One. Linear Systems2.3 Lemma (Linear Combination Lemma) A linear combination of linear combinationsis a linear combination.Proof Given the set c 1,1 x 1 + · · · + c 1,n x n through c m,1 x 1 + · · · + c m,n x n oflinear combinations of the x’s, 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 also a linear combination of the x’s.QED2.4 Corollary Where one matrix reduces to another, each row of the second is alinear combination of the rows of the first.The proof uses induction. ∗ Before we proceed, here is an outline of theargument. For the base step, we will verify that the proposition is true whenreduction can be done in zero row operations. For the inductive step, we willargue that if being able to reduce the first matrix to the second in some numbert 0 of operations implies that each row of the second is a linear combinationof the rows of the first, then being able to reduce the first to the second in t + 1operations implies the same thing. Together these prove the result because thebase step shows that it is true in the zero operations case, and then the inductivestep implies that it is true in the one operation case, and then the inductivestep applied again gives that it is therefore true for two operations, etc.Proof We proceed by induction on the minimum number of row operations thattake a first matrix A to a second one B. In the base step, that zero reductionoperations suffice, the two matrices are equal and each row of B is trivially acombination of A’s rows: ⃗β i = 0 · ⃗α 1 + · · · + 1 · ⃗α i + · · · + 0 · ⃗α m .For the inductive step, assume the inductive hypothesis: with t 0, anymatrix that can be derived from A in t or fewer operations then has rowsthat are linear combinations of A’s rows. Suppose that reducing from A toB requires t + 1 operations. There must be a next-to-last matrix G so thatA −→ · · · −→ G −→ B. The inductive hypothesis applies to this G because it isonly t operations away from A. That is, each row of G is a linear combinationof the rows of A.If the operation taking G to B is a row swap then the rows of B are just therows of G reordered, and thus each row of B is a linear combination of the rowsof G. If the operation taking G to B is multiplication of some row i by a scalar cthen the rows of B are a linear combination of the rows of G; in particular,⃗β i = c⃗γ i . And if the operation is adding a multiple of one row to another then∗ More information on mathematical induction is in the appendix.