By David B. Damiano

ISBN-10: 0155151347

ISBN-13: 9780155151345

The authors outline basic vector areas and linear mappings on the outset and base all next advancements on those suggestions. This strategy presents a ready-made context, motivation, and geometric interpretation for every new computational approach. Proofs and summary problem-solving are brought from the beginning, supplying scholars an instantaneous chance to perform utilising what they have realized. each one bankruptcy comprises an creation, precis, and supplementary workouts. The textual content concludes with a couple of worthy appendixes and strategies to chose exercises.

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I, . , bm) X\(ci\i, . , am\) + 33 + xn(a\fn . , amn) — (« nT| + • • • + ai„xn, . v„ = b, fl21-Yl + ' • ' + C l2 „ X n = b2 am\X\ + • • • + ci„mx„ - b,„ We ask, are there any solutions to the system? „,), what conditions must the fr, satisfy for solutions to exist ? Note that except for the possible nonzero right-hand sides b,. the form of these equations is the same as that of the ones previously considered. Finally, let us consider problems of the third type. ) Again, let V = R'" and let 5 = {(an , .

INTERLUDE ON SOLVING SYSTEMS OF LINEAR EQUATIONS 45 This is a homogeneous system of three equations in the unknowns «,, «2, a3. From this point we may proceed exactly as we would in R3. After reducing the system to echelon form we have «1 — «3 = 0 a2 + 2a3 = 0 0 = 0 Since a3 is a free variable, there are nontrivial solutions, such as (a ,,a 2,«3) = ( 1 ,-2 ,1 ), and this implies that S is linearly dependent. The reader should check that these scalars actually do give us a linear dependence among the three poly nomials in the set S.

Then - l x + x = ( - 1 + l)x = 0x = 0 E W . Now let x be any vector in W and let y = 0. Then cx + 0 = cx G W, so W is closed under scalar multiplication. To see that these observations imply that W is a vector space, note that the axioms 1. 2, and 5 through 8 in Definition ( 1. 1. 1) are satisfied automatically for vectors in W. since they hold for all vectors in V. Axiom 3 is satisfied, since as we have seen 0 G W. 1,6d) ( —I)x = —x G W as well. Hence W is a vector space. ■ To see how the condition of the theorem may be applied, we consider several examples.