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 techniques. This strategy offers a ready-made context, motivation, and geometric interpretation for every new computational process. Proofs and summary problem-solving are brought from the beginning, delivering scholars a right away chance to perform utilizing what they have discovered. every one bankruptcy includes an advent, precis, and supplementary workouts. The textual content concludes with a couple of worthwhile appendixes and strategies to chose exercises.

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**Example text**

12) E xam ple. In V = R2, consider W = {(â,, a 2) | . / — 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.

7) Rem ark. The name “echelon form” comes from the step- or ladder-like pattern of an echelon form system when space is left to indicate zero coefficients and line up the terms containing each variable. 5) the following statement should be extremely plausible. 8) T heorem . Every system of linear equations is equivalent to a system in echelon form. Moreover this echelon form system may be found by applying a sequence of elementary operations to the original system. We will prove the theorem by an important technique we have not used before called mathematical induction.

2 SUBSPACES 19 aux, + a i2x 2 + ■ ■ ■ + a inx„ = 0 « 21*1 + < 122 X2 + • • • + a2lfx„ = 0 am\X1 + am2x 2 + ■ ■ ■ + am„x„ = 0 is a subspace of R". (Here the notation ay means the coefficient of x2 in the ith equation. 14) Corollary. Let a,j ( 1 =£ / =s m, 1 $ ; ' « « ) be any real numbers and let W = {(*,, . . ,x„) E R" | anx 1 + • • ■ + ainx„ = 0 for all /, 1 =£ i =s m } . Then W is a subspace of R". Proof: For each/, 1 =£ / =¾ m, let W, = {(x1; . . j n) \ anx, + • ■ • + a,„x„ = 0}. Then since W is precisely the set of solutions of the simultaneous system formed from the defining equations of all the Wn we have W = Wt D W2 D • • ■ n Wm.

### A Course in Linear Algebra by David B. Damiano

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