By Philippe Dennery
First-class textual content offers thorough heritage in arithmetic had to comprehend state-of-the-art extra complicated themes in physics and engineering. subject matters comprise concept of features of a fancy variable, linear vector areas, tensor calculus, Fourier sequence and transforms, unique services, extra. Rigorous theoretical improvement; difficulties solved in nice aspect. Bibliography. 1967 version.
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Extra resources for Mathematics for Physicists (Dover Books on Mathematics)
Example text
N) sind, hei:6t eine (m, n)-Nullmatrix (oder: Nullmatrix 0). Unter der transponierten Matrix (oder: Transponierten) AT einer (m,n)Matrix A= am 2 ... versteht man die (n, m)- Matrix an a2I . " a m2 amn' Sie entsteht folglich durch Vertauschung von Zeilen und Spalten von A. Beispiel: A = ( 21) ~ ~ {:::::} AT = (230) 1 2 6 . ;~ hei:6t quadratisch, falls m = n (Zeilenanzahl = Spaltenanzahl) gilt. 1st dariiber hinaus aik = aki fiir alle i, k = 1, ... , m = n, d. h. A = AT, so hei:6t A symmetrisch.
N) aus. Dabei konnen folgende Fane eintreten: Fall 1: Alle Yi(i = 1, ... ,m) lassen sich gegen gewisse xk(k = 1, ... ,n) austauschen. Fall 2: Mindestens ein Yu ist nicht mehr gegen ein kann dann mindestens eine Abhangigkeitsbeziehung Yu = f,L1Y1 + f,L2Y2 + ... + f,LmYm + f,Lo Xk austauschbar. Man (A) ablesen, wobei auf der rechten Seite von (A) natiirlich hOchstens die ausgetauschten Yi vorkommen. Dabei sind wieder zwei Fane moglich: = O. Fall 2b: In (A) ist f,Lo '" o. Fall 2a: In (A) ist f,Lo Das lineare Gleichungssystem (G) ist also genau dann losbar, wenn der Fall 2b nicht eintritt.
K=l, ... ,n a,k i=l, ... ,m Kapitel 2 Lineare Algebra und Optimierung 44 mittels ihrer m . n Koeffizienten (oder: Elemente) aile (i = 1, ... ,mj k = 1, ... , n) angegeben. Als Spalten (oder: Spaltenvektoren) bezeichnet man die Vektoren (an aI n ) analog ala Zeilen (oder: Zeilenvektoren) von A. t = 1, ... , mj k = · M a trIX Eme i=I ..... Zlent en aik = 1, ... , n) sind, hei:6t eine (m, n)-Nullmatrix (oder: Nullmatrix 0). Unter der transponierten Matrix (oder: Transponierten) AT einer (m,n)Matrix A= am 2 ...