Let V be the set V = {(x, 1, z) | x, z E R}. Define the addition e and scalar multiplication Oon V by(x, 1, 2) ® (x', 1, z') =(x + x', 1, z + 2'),kO (x, 1, z) =(ka, 1, kz),k ERDetermine whether the set V with addition and scalar multiplication © is a vector space.Justify your answers.

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Answered: Let V be the set V = {(x, 1, z)| x, z E…

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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Rather than use the standard definitions of addition and scalar multiplication in R, suppose these two operations are defined as follows. With these new definitions, is Ra vector space? ustify your answers. () ( Y. ) - 2) - * *. n* * 2) x. Y. 2) - (ox, cy, 0) O The set is a vector space. O The set is not a vector space because the associative property of addition is not satisfied. O The set is not a vector space because it is not closed under scalar multiplication. O The set is not a vector space because the associative property of multiplication is not satisfied. O The set is not a vector space because the multiplicative identity property is not satisfied. (6) . Ya. 72) - . V 2) = (0, 0. 0) c, Y. 2) - (ox, y, cz) O The set is a vector space. O The set is not a vector space because the commutative property of addition is not satisfied. O The set is not a vector space because the additive identity property is not satisfied. O The set is not a vector space because it is not closed under…
6 Let (u, v) be the inner product on R² generated by A = [ −1 and let u = (3,5), v = (7, 4). Find (u, v). (u, v) 1 -2 3
Consider the set R²={(a,b)Da,b ER}. Suppose the following operations are defined on this set. Addition: (X 1.y 1)® (x 2Y2)=(x1+X2.Y1+Y2) • Scalar Multiplication: C © (x.y)=(cx,÷y) Which of the following axioms for a vector space fail to hold under these operations? O Distributivity of scalar multiplication over vector addition (c +d)O u=(c ©u)e (dO u) for all scalars C,d and all ordered pairs u=(x.y) 10 u=u for all u ER? O Closure under addition Additive identity QUESTION 15 Find the coordinates of X= (13, 26, 39) relative to the orthonormal basis B = inR 13 49 29 -78 O b.(x], =| 58 39 61 =| 58
22. Show that there do not exist scalars c₁, c₂, and c² such that c₁(1,0, 1,0) + c₂(1, 0, −2, 1) + c3(2, 0, 1, 2) = (1, -2, 2, 3)
Suppose that ū, ī and w are vectors in a real inner product space such that (7, w) = 16, (T, w) = 5, d(2ū, v) = v27 and ||W|| = V27. Show that 27 - 0 - w = 0.
3. (12) Let V = R². Define vector addition and scalar multiplication as follows: (u1, uz)O(v1, v2) = (u1V1, U2V2) and c(u1,u2) = (u,C, u2^), c > 0 (a) Does V have an additive identity? If so, what would it be? (b) Does the distributive axiom c(uOv) = cu@cv hold? (Show why or why not). 4
"On R², define the operations of addition and scalar multiplication as follows: (X₁, X₂) + (y₁ Y₂) = (x₂ + y₂ +1, x₂ + y₂ + 1), c(x₁, x₂) = (Cx₂ + C-1, ₂+c-1). Then R² with these operations forms a vector space." Note that here the zero vector is (-1,-1) and the additive inverse of (x1, x2) is (-X₁-2, -x₂-2), not (-x₂-x₂)- a) Show that (-1, -1) acts as the zero vector under these definitions.
(2) Let aj = (1,2, -1), a2 = (3,0, 1) and bị = (-1, 10, 7), b2 = (1, 2, 0) in R³. %3D (a) Is a, in Span(a,)? (b) Which of bị, b2 is in Span(a, a2)? (c) Write each b; in Span(a1, a2) as a linear combination of a1, a2.
The set containing the elements specified also possesses its relevant operations of multiplication over reals and vector addition. Is the set a real vector space?

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Let V be the set V = {(x, 1, z)| x, z E R}. Define the addition and scalar multiplication on V by (x, 1, 2) (x', 1, 2') =(x+ x', 1, z + 2'), kO (x, 1, z) =(kx, 1, kz), kER Determine whether the set V with addition and scalar multiplication is a vector space. Justify your answers.