Problem and addition () by s] Let R+ be the set of positive real numbers. Define scalar multiplication (o) ax = xa for all x R+, α = R xy=xy, for all x, y = R+ -4 = 16, and the addition of 3 For example, the scalar product of -4 times ½ is -4 0 1 = (})¯ and 7 is 37=3.7 = 21. 5) Considering the operations and H, is R+ a vector space? Yes No. Justify your answer (briefly explain why each of the axioms A1 to A8 hold or fail) (x = − z = x + (y = z). A1 x y = y = x. A2 A3 x0 =x=0x. A4 xinv(x) = 0. A5 A6 a ○ (x = y) = a ° x = a ○ y. (a+b)ox=aox Box. A7 ao (Box) = (aẞß) ox. A8 10x = x. Commutativity of addition. Associativity of addition Exists a neutral element for add. Every x has an inverse inv(x). Vectors distribute. Scalars distribute. Associativity of scalar multiplication. Scaling by 1 gives the same.
Problem and addition () by s] Let R+ be the set of positive real numbers. Define scalar multiplication (o) ax = xa for all x R+, α = R xy=xy, for all x, y = R+ -4 = 16, and the addition of 3 For example, the scalar product of -4 times ½ is -4 0 1 = (})¯ and 7 is 37=3.7 = 21. 5) Considering the operations and H, is R+ a vector space? Yes No. Justify your answer (briefly explain why each of the axioms A1 to A8 hold or fail) (x = − z = x + (y = z). A1 x y = y = x. A2 A3 x0 =x=0x. A4 xinv(x) = 0. A5 A6 a ○ (x = y) = a ° x = a ○ y. (a+b)ox=aox Box. A7 ao (Box) = (aẞß) ox. A8 10x = x. Commutativity of addition. Associativity of addition Exists a neutral element for add. Every x has an inverse inv(x). Vectors distribute. Scalars distribute. Associativity of scalar multiplication. Scaling by 1 gives the same.
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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![Problem
and addition () by
s] Let R+ be the set of positive real numbers. Define scalar multiplication (o)
ax = xa
for all x R+, α = R
xy=xy,
for all x, y = R+
-4
=
16, and the addition of 3
For example, the scalar product of -4 times ½ is -4 0 1 = (})¯
and 7 is 37=3.7 = 21.
5) Considering the operations and H, is R+ a vector space?
Yes No.
Justify your answer (briefly explain why each of the axioms A1 to A8 hold or fail)
(x =
− z = x + (y = z).
A1
x y
=
y = x.
A2
A3
x0 =x=0x.
A4 xinv(x) = 0.
A5
A6
a ○ (x = y) = a ° x = a ○ y.
(a+b)ox=aox
Box.
A7
ao (Box) = (aẞß) ox.
A8
10x = x.
Commutativity of addition.
Associativity of addition
Exists a neutral element for add.
Every x has an inverse inv(x).
Vectors distribute.
Scalars distribute.
Associativity of scalar multiplication.
Scaling by 1 gives the same.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feb22fec7-de0b-4f6c-8978-5553e85e3b14%2F43718033-3af9-4f6f-97cb-38fccdab5be4%2F618lnuy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem
and addition () by
s] Let R+ be the set of positive real numbers. Define scalar multiplication (o)
ax = xa
for all x R+, α = R
xy=xy,
for all x, y = R+
-4
=
16, and the addition of 3
For example, the scalar product of -4 times ½ is -4 0 1 = (})¯
and 7 is 37=3.7 = 21.
5) Considering the operations and H, is R+ a vector space?
Yes No.
Justify your answer (briefly explain why each of the axioms A1 to A8 hold or fail)
(x =
− z = x + (y = z).
A1
x y
=
y = x.
A2
A3
x0 =x=0x.
A4 xinv(x) = 0.
A5
A6
a ○ (x = y) = a ° x = a ○ y.
(a+b)ox=aox
Box.
A7
ao (Box) = (aẞß) ox.
A8
10x = x.
Commutativity of addition.
Associativity of addition
Exists a neutral element for add.
Every x has an inverse inv(x).
Vectors distribute.
Scalars distribute.
Associativity of scalar multiplication.
Scaling by 1 gives the same.
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