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9. Let V be a set consisting of a single element z. De- fine addition and scalar multiplication on V by 'z = 2+ 2 2 = 2) Show that V is a vector space. Such a vector space is called a zero vector space.

9. Let V be a set consisting of a single element z. De- fine addition and scalar multiplication on V by 'z = 2+ 2 2 = 2) Show that V is a vector space. Such a vector space is called a zero vector space.

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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0. numbers Ln-=1 an. Is S a vector space under the convergent series of real addition and scalar multiplication 00 E an +E bn = (an + bn), n=1 n=1 n=1 Σ Eca,? An = n=1 n=1 real If not, why not? and 9. Let V be a set consisting of a single element z. De- fine addition and scalar multiplication on V by y^). 'z = 2+ 2 Cz = z. Show that V is a vector space. Such a vector space ition is called a zero vector space. 10. Prove part (2) of Theorem 2.2. 11. Prove that if c is a real number and v is a vector in a vector space V such that cv = 0, then either c = 0 or v = 0. wector ces of ler the 12. Show that subtraction is not an associative operation on a vector space. NNING SETS - with subspaces. Roughly speaking, by a subspace vithin a larger vector space. The following definition states u we mean a A subset W of a vector space V is cailed a subspace of V if W -pace under the addition and scalar mu ion of V restricted all column vectors of the
Transcribed Image Text:0. numbers Ln-=1 an. Is S a vector space under the convergent series of real addition and scalar multiplication 00 E an +E bn = (an + bn), n=1 n=1 n=1 Σ Eca,? An = n=1 n=1 real If not, why not? and 9. Let V be a set consisting of a single element z. De- fine addition and scalar multiplication on V by y^). 'z = 2+ 2 Cz = z. Show that V is a vector space. Such a vector space ition is called a zero vector space. 10. Prove part (2) of Theorem 2.2. 11. Prove that if c is a real number and v is a vector in a vector space V such that cv = 0, then either c = 0 or v = 0. wector ces of ler the 12. Show that subtraction is not an associative operation on a vector space. NNING SETS - with subspaces. Roughly speaking, by a subspace vithin a larger vector space. The following definition states u we mean a A subset W of a vector space V is cailed a subspace of V if W -pace under the addition and scalar mu ion of V restricted all column vectors of the
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