Let V be the set of functions f: R → R. For any two functions f, g in V, define the sum f + g to be the function given by (ƒ + g)(x) = f(x) + g(x) for all real numbers x. For any real number c and any function f in V, define scalar multiplication of by (cf)(x) = cf(x) for all real numbers x. Answer the following questions as partial verification that V is a vector space. (Addition is commutative:) Let f and g be any vectors in V. Then f(x) + g(x) = for all real numbers x since adding the real numbers f(x) and g(x) is a commutative operation. (A zero vector exists:) The zero vector in V is the function ƒ given by ƒ(x) = for all . (Additive inverses exist:) The additive inverse of the function f in V is a function g that satisfies f(x) + g(x) = 0 for all real numbers. The additive inverse of f is the function g(x) = for all c. (Scalar multiplication distributes over vector addition:) If c is any real number and fand g are two vectors in V then c(ƒ + g)(x) = c(f(x) + g(x)) = ¯ I

Let V be the set of functions f: R → R. For any two functions f, g in V, define the sum f + g to be the function given by (ƒ + g)(x) = f(x) + g(x) for all real numbers x. For any real number c and any function f in V, define scalar multiplication of by (cf)(x) = cf(x) for all real numbers x. Answer the following questions as partial verification that V is a vector space. (Addition is commutative:) Let f and g be any vectors in V. Then f(x) + g(x) = for all real numbers x since adding the real numbers f(x) and g(x) is a commutative operation. (A zero vector exists:) The zero vector in V is the function ƒ given by ƒ(x) = for all . (Additive inverses exist:) The additive inverse of the function f in V is a function g that satisfies f(x) + g(x) = 0 for all real numbers. The additive inverse of f is the function g(x) = for all c. (Scalar multiplication distributes over vector addition:) If c is any real number and fand g are two vectors in V then c(ƒ + g)(x) = c(f(x) + g(x)) = ¯ I

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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Question

A vector space over R is a set V of objects (called vectors) together with two operations, addition and multiplication by scalars (real numbers), that satisfy the following 10 axioms. The axioms must hold for all vectors u, v, win V and for all scalars a, B in R.

Let V be the set of functions f : R → R. For any two
functions f, g in V, define the sum f + g to be the
function given by (ƒ + g)(x) = f(x) + g(x) for all
real numbers x. For any real number c and any function
f in V, define scalar multiplication of by
(cf)(x) = cf(x) for all real numbers x.
Answer the following questions as partial verification
that V is a vector space.
(Addition is commutative:) Let f and g be any vectors
in V. Then f(x) + g(x) = for all real numbers x
since adding the real numbers f(x) and g(x) is a
commutative operation.
(A zero vector exists:) The zero vector in V is the
function ƒ given by ƒ(x) =
for all x.
(Additive inverses exist:) The additive inverse of the
function f in V is a function g that satisfies
f(x) + g(x) = 0 for all real numbers. The additive
inverse of f is the function g(x) = for all c.
(Scalar multiplication distributes over vector addition:)
If c is any real number and fand g are two vectors in V
then c(ƒ + g)(x) = c(ƒ(x) + g(x)) =■
I

1. (Closed under addition:) The sum of u and v,
denoted u + v, is in V.
2. (Closed under scalar multiplication:) The scalar
multiple of u by a, denoted au, is in V.
3. (Addition is commutative:) u + v = v + u.
4. (Addition is associative:)
(u + v) + w =u+ (v+w).
5. (A zero vector exists:) There exists a vector 0 in
V such that u + 0 = u.
6. (Additive inverses exist:) For each u in V, there
exists a v in V such that u + v = 0. (We write
v=-u.)
7. (Scaling by 1 is the identity:) lu = u.
8. (Scalar multiplication is associative):
a(ßu) = (aß)u.
9. (Scalar multiplication distributes over vector
addition:) a (u + v) = au + av.
10. (Scalar addition is distributive:)
(a + B)u = au + Bu.

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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