A Course in Linear Algebra by David B. Damiano

By David B. Damiano

Suitable for complicated undergraduates and graduate scholars, this article deals a whole advent to the fundamental techniques of linear algebra. attention-grabbing and encouraging in its technique, it imparts an knowing of the subject's logical constitution in addition to the ways that linear algebra presents options to difficulties in lots of branches of mathematics.
The authors outline basic vector areas and linear mappings on the outset and base all next advancements on those ideas. This process presents a ready-made context, motivation, and geometric interpretation for every new computational approach. Proofs and summary problem-solving are brought from the beginning, providing scholars a right away chance to perform utilizing what they have discovered. each one bankruptcy includes an advent, precis, and supplementary workouts. The textual content concludes with a couple of precious appendixes and options to chose exercises.

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Why or why not? c) Is always linearly independent? Why or why not? 3. Repeat the three parts of Question 2, replacing linearly independent by linearly dependent. 4. a) Show that if v, w E V, then {v,w} is linearly dependent if an only if v is a scalar multiple of w, or w is a scalar multiple of v. b) Show by example, however, that there are linearly dependent sets of three vectors such that no pair are scalar multiples of each other. 5. Let v, w e V. Show that {v,w} is linearly independent if and only if {v + w, v — w} is linearly independent.

If this is 3= 1, then there are nontrivial solutions of the system, so the original set of vectors was linearly dependent. Otherwise, the set is independent. 15) Exam ple. We work with another vector space V = P„(R) and show that determining if a set of polyomials is linearly independent or not may be accomplished by the same techniques we have discussed in R". Consider the set S = {x2 + 2x + 1, x2 + 4x + 3, x2 + 6x + 5} C P2(R). To determine if 5 is linearly independent, we consider a potential linear dependence «,(x 2 + 2x + 1) + a2(x2 + 4x + 3) + a3(x2 + 6* + 5) = Ox2 + Ox + 0, or, after collecting like terms in x, (a, + a2 + a3)x2 + (2 a, + 4 a2 + 6a3)x + (a, + 3a2 + 5a3) = Ox2 + Ox + 0.

X \ = 0}. 1 ) and (4, 8) £ W, but the sum (1, 1) + (4, 8) = (5. 9) £ W. The components of the sum do not satisfy the defining equation of W: 53 - 92 = 44 =£ 0. 10). , = 0} then W is the set of vectors in both and W2. In other words, we have an equality of sets W = W t Ci W2. Inthis example we see that the intersection of these two subspaces of R 3 is also a subspace of R \ This is a property of intersections of subspaces, which is true in general. 13) T heorem . Let V be a vector space. Then the intersection of any col­ lection of subspaces of V is a subspace of V.

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