A circuit is a closed loop, ig,

A circuit is a closed loop, ig,....%n—1, 88 = to of length n > 0. Conservation

of energy is a condition on the & x & matrix V to the effect that, on cach circuit the sum is zero

Vio, in) + V (in, 8a) + +++ + Vina, io) = 0.

A matrix satisfying this condition ix called conservative. Show that cach conservative matrix is shew- symmetric. Deduce that the set of conservative matrices is a vector space, closed under vector-space op-erations, Exhibit a 3 x 3 skew-symmetric matrix that is not conservative. A skew-symmetric matrix of the form V(i,j) — a, — a, ix called additive. Prove that every conservative matrix is additive. What is the dimension of the vector space of conservative 6 x 6 matrices?

The following exercises refer to the linear model for the voltages in which E(Y,,) — a; — a, is conservative.

The data is ax follows:

Electrolyte O

Mg Zu Fe Ph Cu

Mg 0.0 O41 0.807 Os7é 1.298

Za ~O44 0.0 042 0.533 0.886

Fe -0.807 -0.429 0 0.043 0377

Pb ~ 0.876 0533 HS oo a7

cu 0.291 0.886 -O.377 -0.271 oo

Electrolyte A

Mg 0.0 0.217 0.56 1.051 1.002

Zo -O0.247 0.0 OU 0.521 0.867

Fe 0.856 ~O.4 00 0.058 O.443

Pb -1.051 0.521 0.058 0.0 0.374

Cu — 1.402 0.867 048 0.374 0.0

Electrolyte K

Mg 00 0.443 0.895 0.973 1.281

Zn -O0.443 00 OAT? 0.503 0.856

Fe -0.885 ~O.AT7 00 0.107 0.432

Pb 0.973 -0.508 -0.107 00 0.392

cu ~ Last 0 356 ~ 0432 —0,592 ao

{a) For a single & x & table, obtain an expression for the least squares estimate of a = (01,...,05). Use this formula to compute & for each of the three electrolytes. Explain why (a1,....05) and (a1,....a5) +

{e,¢.¢,¢,¢) are equivalent as parameter points in the model.

{b) Assess the evidence for and against the hypothesis that the vector of potentials is constant across elec-

trolytes. That is to say, fit the linear model in which the potentials are constant across clectrolytes, and 5 compare the fit with the model in which a varies from one electrolyte to another, Obtain the relevant Act sums of squares, their degrees of freedom, and compute the appropriate F statistic. od ti

(c) Discuss brietly the arguments for and against analysis of these data by lincar models after transformation.