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2. Determine whether the following sets with the corresponding vector addi- tion and scalar multiplication form vector spaces. a) Let V={(₁,₂): ₁, ₂ R). For (a₁,₂), (b₁,b₂) EV and e € R, define: (a₁.a₂)+(b₁,b₂)(a1 +2b₁,02+36₂) and c(a₁.₂) (ca.co₂) b) Let V=((a.az): 01.02 R). For (@a₁,0₂) V and c R, define: c(0₁.0₂) = (ca₂.0)

2. Determine whether the following sets with the corresponding vector addi- tion and scalar multiplication form vector spaces. a) Let V={(₁,₂): ₁, ₂ R). For (a₁,₂), (b₁,b₂) EV and e € R, define: (a₁.a₂)+(b₁,b₂)(a1 +2b₁,02+36₂) and c(a₁.₂) (ca.co₂) b) Let V=((a.az): 01.02 R). For (@a₁,0₂) V and c R, define: c(0₁.0₂) = (ca₂.0)

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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2. Determine whether the following sets with the corresponding vector addi- tion and scalar multiplication form vector spaces. a) Let V={(₁,₂): ₁, ₂ R). For (a,,a2), (b₁,b₂) EV and e R, define: (a₁.ay)+(b₁,b₂)=(a₁ +2b₁,02+36₂) and c(a₁.az) (ca.co₂) b) Let V=((a.az): 01.02 € R). For (1,0₂) € Vand c c(a.az) = (ca₁,0) Step 1: Definition of vector space A mon-empty set V is said to form a vector ever a field F if stence (1) there is a binary composition + on V, called addition, satisfying the conditions 1. d+pe V for all diper 2. R+B=p+d for all di ptv. 1. α+ (1+x) = (x+1)+d for all *₁1₁*6V. 4. there exists an element in Vluch that 1. O Solution , define: Step 2: Check the properties x+D = α for all dev 5. for each af in V there exists an element no in V such that and (1) there is an external composition of F with V, called 'multiplicationly the elements of F' satisfying the conditions, 6. cat v for all c&f, all dev. 7. c(da) = (cd) α for all lide F all dev 8. C(R+p) = cateß for all off all Afv 9. (c+d) α = ca+dx for all of V, code F 10₁ 1α = α, 1 being the identity element inf. 0, the Here given V = {(9₁, 9₂): 9₁, 9₂ ER}. for (0,0₂), (b₁,b₂) EV and CEIR, define: (o₁, A₂)+ (b₁,b₂) = (a₁ +26₁, A₂+36₂) and c(0₁, as) = (cay, ca₂). het, (0, ₂), (b₁,b₂), (C₁, C₂) EV and CidER. (ay, as) + (bist₂) = (04+26₁, A₂+at₂) EV (ou, as) + (bi, b₂) = (a₁ +26, ₂+ab₂) (b₁,b₂) + (ay, a₂) = (by+201, b₂+ 39₂), So, (ar, n) + (b₁,b₂) + (t₁,b₂) + (a₁, 92.) Exemple, Consider, (1,1), (2,2) EV. (1,1)+(2₁2) = (1+4, 1+6) = (5,7) (212) + (1₁1) = (2+2, 2+3) = (4.5). ..V is not a vector space with respect to addition and scalar multiplication.
Transcribed Image Text:2. Determine whether the following sets with the corresponding vector addi- tion and scalar multiplication form vector spaces. a) Let V={(₁,₂): ₁, ₂ R). For (a,,a2), (b₁,b₂) EV and e R, define: (a₁.ay)+(b₁,b₂)=(a₁ +2b₁,02+36₂) and c(a₁.az) (ca.co₂) b) Let V=((a.az): 01.02 € R). For (1,0₂) € Vand c c(a.az) = (ca₁,0) Step 1: Definition of vector space A mon-empty set V is said to form a vector ever a field F if stence (1) there is a binary composition + on V, called addition, satisfying the conditions 1. d+pe V for all diper 2. R+B=p+d for all di ptv. 1. α+ (1+x) = (x+1)+d for all *₁1₁*6V. 4. there exists an element in Vluch that 1. O Solution , define: Step 2: Check the properties x+D = α for all dev 5. for each af in V there exists an element no in V such that and (1) there is an external composition of F with V, called 'multiplicationly the elements of F' satisfying the conditions, 6. cat v for all c&f, all dev. 7. c(da) = (cd) α for all lide F all dev 8. C(R+p) = cateß for all off all Afv 9. (c+d) α = ca+dx for all of V, code F 10₁ 1α = α, 1 being the identity element inf. 0, the Here given V = {(9₁, 9₂): 9₁, 9₂ ER}. for (0,0₂), (b₁,b₂) EV and CEIR, define: (o₁, A₂)+ (b₁,b₂) = (a₁ +26₁, A₂+36₂) and c(0₁, as) = (cay, ca₂). het, (0, ₂), (b₁,b₂), (C₁, C₂) EV and CidER. (ay, as) + (bist₂) = (04+26₁, A₂+at₂) EV (ou, as) + (bi, b₂) = (a₁ +26, ₂+ab₂) (b₁,b₂) + (ay, a₂) = (by+201, b₂+ 39₂), So, (ar, n) + (b₁,b₂) + (t₁,b₂) + (a₁, 92.) Exemple, Consider, (1,1), (2,2) EV. (1,1)+(2₁2) = (1+4, 1+6) = (5,7) (212) + (1₁1) = (2+2, 2+3) = (4.5). ..V is not a vector space with respect to addition and scalar multiplication.
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