Problem 26E

As an alternative proof that the functions er1x,er2x, . . . , ernx are linearly independent on when r1,r2 . . . . ,rn are distinct, assume holds for all x in and proceed as follows:

(a) Because the ri’s are distinct we can (if necessary) relabel them so that r1 > r2 > . . . > rn. Divide equation (33) by er1x to obtain Now let on the left-hand side to Obtain C1=0

(b) Since C1=0, equation (33) becomes for all x in Divide this equation by er2x and let to conclude that C2=0

(c) Continuing in the manner of (b), argue that all the coefficients, C1, C2, . . . , Cn are zero and hence er1x,er2x, . . . ,erex are linearly independent on

Chapter 9 from “Introductory Chemistry” by Zumdahl and Decoste, 7 /8 editionth Pg 203. INFORMATION GIVEN BY CHEMICAL EQUATIONS *Coefficient in a balanced equation give us the relevant number of molecules Note: we need to focus on the ratio of coefficients not the specific coefficients themselves Ex: The ratio of item when making a sandwich is: 2 bread slices + 3 meat slices + 1 cheese slice = 1 sandwich Ratio 2:3:1 *and no matter what number you multiply the equation by, the ratio is always 2:3:1 Pg 205. MOLE TO MOLE RELATIONSHIPS -We use equations to predict how many moles of products a # of moles of reactants will produce Ex: the decomposition of water into H and O 2H 2 (l) 2H (2) + O (g2 2 moles of water (H 2) gives 2 moles of hydrogen (H ) a2d 1 mole of oxygen (O )