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ctor space? If not, why not? 7. Let C denote the set of all convergent sequences of real numbers {an}. Is C a vector space under the

ctor space? If not, why not? 7. Let C denote the set of all convergent sequences of real numbers {an}. Is C a vector space under the

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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triples of real numbers yl those that are not vector spaces, determine which properties of a vector space fail to hold. a) (x1, y1, Z1) + (x2, y2, Z2) = (x1 + x2, y1 + y2, Z1 + z2), c(x, y, z) = (cx, y, cz) b) (x1, yı, Z1) + (x2, Y2, Z2) = (Zi + 2, yı + y2, X1 +x2), с(х, у, г) %3D (сх, су, сг) c) (x1, yı, Z1) + (x2, Y2, 22) = (x + x2, y1 + y2 - 2, z1 + z2), c(x, y, z) = (cx, y, z) 5. Show that the set of ordered pairs of positive real numbers is a vector space under the addition and scalar multiplication If not, why n 8. Let S denote numbers addition and %3D Σ n=1 If not, why ne 9. Let V be a se fine addition (x1, yı)+(x2, y2) = (x1X2, Y1Y2), c(x, y) = (x, y^). 6. Does the set of complex numbers under the addition and scalar multiplication Show that V is called a zer (a + bi) + (c + di) = (a + c) + (b + d)i, 10. Prove part (2) c(a + bi) = ca + cbi 11. Prove that if c where a, b, c, and d are real numbers form a vector space? If not, why not? 7. Let C denote the set of all convergent sequences of real numbers {an}. Is C a vector space under the vector space or v = 0. 12. Show that subt on a vector sp= 2.2 SUBSPACES AND SPANNING SETS We begin this section with subspaces. Roughly sp vector space sitting within a larger vector space. precisely. DEFINITION A subset W of a vector space
Transcribed Image Text:triples of real numbers yl those that are not vector spaces, determine which properties of a vector space fail to hold. a) (x1, y1, Z1) + (x2, y2, Z2) = (x1 + x2, y1 + y2, Z1 + z2), c(x, y, z) = (cx, y, cz) b) (x1, yı, Z1) + (x2, Y2, Z2) = (Zi + 2, yı + y2, X1 +x2), с(х, у, г) %3D (сх, су, сг) c) (x1, yı, Z1) + (x2, Y2, 22) = (x + x2, y1 + y2 - 2, z1 + z2), c(x, y, z) = (cx, y, z) 5. Show that the set of ordered pairs of positive real numbers is a vector space under the addition and scalar multiplication If not, why n 8. Let S denote numbers addition and %3D Σ n=1 If not, why ne 9. Let V be a se fine addition (x1, yı)+(x2, y2) = (x1X2, Y1Y2), c(x, y) = (x, y^). 6. Does the set of complex numbers under the addition and scalar multiplication Show that V is called a zer (a + bi) + (c + di) = (a + c) + (b + d)i, 10. Prove part (2) c(a + bi) = ca + cbi 11. Prove that if c where a, b, c, and d are real numbers form a vector space? If not, why not? 7. Let C denote the set of all convergent sequences of real numbers {an}. Is C a vector space under the vector space or v = 0. 12. Show that subt on a vector sp= 2.2 SUBSPACES AND SPANNING SETS We begin this section with subspaces. Roughly sp vector space sitting within a larger vector space. precisely. DEFINITION A subset W of a vector space
10. Prove part (2) of Theorem 2.2. 9. Let V be a set consisting of a single element z. 8. Let S denote the set of all convergent series of re {an} + {bn} = {an + bn}, c{an} = {ca,? addition and scalar multiplication r the indi- f ordered асе. For ne which If not, why not? numbers n=i an. Is S a vector space under addition and scalar multiplication 00 Ean + bn = (an + b.). n=1 n=1 n=1 00 2an = ca,? C n=1 n=1 ive real ion and If not, why not? fine addition and scalar multiplication on V =*, y°). z+z = z, ddition is called a zero vector space. )i, vector or v = 0. aces of
Transcribed Image Text:10. Prove part (2) of Theorem 2.2. 9. Let V be a set consisting of a single element z. 8. Let S denote the set of all convergent series of re {an} + {bn} = {an + bn}, c{an} = {ca,? addition and scalar multiplication r the indi- f ordered асе. For ne which If not, why not? numbers n=i an. Is S a vector space under addition and scalar multiplication 00 Ean + bn = (an + b.). n=1 n=1 n=1 00 2an = ca,? C n=1 n=1 ive real ion and If not, why not? fine addition and scalar multiplication on V =*, y°). z+z = z, ddition is called a zero vector space. )i, vector or v = 0. aces of
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