By P. M. Cohn (auth.)
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Der zweite Band der linearen Algebra führt den mit "Lineare Algebra 1" und der "Einführung in die Algebra" begonnenen Kurs dieses Gegenstandes weiter und schliesst ihn weitgehend ab. Hierzu gehört die Theorie der sesquilinearen und quadratischen Formen sowie der unitären und euklidischen Vektorräume in Kapitel III.
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Additional resources for Elements of Linear Algebra
If the third equation of our system had been 4x 2 - 5X3 = 3, then the same reduction would lead to 1 o o 5 2 9 -4 5 0 0 -3 0 The row of zeros corresponds to the equation 0 = 0, which may be omitted without affecting the solution, so we are left with 2 equations in 3 unknowns. This system cannot have a unique solution, because we can assign X 3 arbitrarily and then solve for Xl' X 2 as before; therefore the system cannot be regular. R2 -3 I R4~R4+R3 I -2 -4 From the triangular form found we obtain 4x 4 = - 4, hence X4 = - 1, x 3 = - 2 - 2x 4 = 0, X2 = -1- 2X3 + 4x 4 = - 41, Xl = 5 + 2x 2 + 7X3 - 4x 4 = 3, so the unique solution is x = (3, - 41,0, - 1)T.
Xnen) = A(xle l ) + ... + A(xne n) = xlAe l + ... 4). Thus we have expressed Ax as a linear combination ofthe n vectors Ae l , ... , Aen, which of course do not depend on the choice ofx. These vectors Aej are nothing but the columns of A: Aej = (alj' a2j, ... , amj)T. 3 EQUALITY OF MATRICES Two matrices A and Bare said to be equal, A = B, if they have the same number of rows and columns - say, they are both m x n matrices - and if corresponding entries are equal: aij = bij (i = 1, ... , m, j = 1, ...
E. b;'ib;'j. Suppose first that i # j and consider the Ath entry in this sum, corresponding to the Ath edge. It is 1 if the edge joins Pi and Pj' and 0 otherwise, so in all the total is just 1 if there is an edge joining Pi and Pj' and 0 otherwise, and this is just aij . b~i' which is the number of edges ending at Pi and this is just the order di at Pi' while aii = 0, by hypo thesis. 27). To illustrate the result, we have for the incidence matrices of the graphs in Figs. ~} (: ~ l) and for the products BTB we obtain The diagonal terms give the order matrices, while the non-diagonal terms give the adjacency matrices.