Linear Algebra and Multi-Dimensional Geometry

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LINEAR ALGEBRA AND MULTI DIMENSIONAL GEOMETRY M IR P U B L I S H E R S

N. V. EFIMOV E. R. ROZENDORN

•

MOSCOW

NV EFIMOV

LINEAR

E.R.ROZENDORN

ALGEBRA

AND MULTI­ DIMENSIONAL GEOMETRY

H. B. EHMOB 3.

P. P03EH JJ0PH

J1HHEHHAH

AJFEBPA H MHOrOMEPHAH TEOMETPMfl

H3AATEJIbCTBO cHAyKA»

N. V. IT IMOV E. R. ROZENDORN

LINEAR ALGEBRA AND MULTI­ DIMENSIONAL GEOMETRY Translated from the Russian by GEORGE YANKOVSKY

MI R

P U B L I S H E R S - M O S C O W

First published 1975 Revised from the 1974 Russian edition

H a aMAUtiCKOM R3blKe

(C) II i;wito;u.ctbo «IlayKa», 1974 (C) IIukIInIi translation, Mir Publishers, 1975

CONTENTS

I*i ••fiii n .................................................................................................................... lull • • • >x be arbitrary scalars. Definition 1. Any element x of the space L that can be repre­ sented as x = aa + pb + yc + . . . -f xq is called a linear combination of the elements a,b,c, . . . , q. We also say that x is expressed linearly in terms of a, b, c........ q. Definition 2. A linear combination is termed trivial if a = p = = Y = . . . = x = 0 and nontrivial if there is at least one nonzero scalar among the scalars a, p, . . . , x. Definition 3. A system (set) of vectors a,b,c, ... . q is said to be linearly dependent if there is a nontrivial linear combination of vectors a, b, c, . . . , q equal to the zero vector, in other words, if it is true that aa + pb + yc + • • • + v.q — 0 where there is at least one nonzero scalar among the scalars «, P, Y........ x-

20

LINEAR SPACES

(CH. I

Definition 4. A system of vectors a, ft, c, . .., q is said to be linearly independent if the equation aa + + yc + • • - + x