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 uv 0. (We write v = -u.) 7. (Scaling by 1 is the identity:) lu = u. 8. (Scalar multiplication is associative): a(Bu) = (aß)u. 9. (Scalar multiplication distributes over vector addition:) a(u+v) = au +αν. 10. (Scalar addition is distributive:) (a + B)u = au + Bu. 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 (f+9)(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 f given by f(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 x. The additive inverse of f is the function g(x)= for all x. (Scalar multiplication distributes over vector addition:) If c is any real number and f and g are two vectors in V, then c(f+9)(x) = c(f(x) + g(x)) =
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 uv 0. (We write v = -u.) 7. (Scaling by 1 is the identity:) lu = u. 8. (Scalar multiplication is associative): a(Bu) = (aß)u. 9. (Scalar multiplication distributes over vector addition:) a(u+v) = au +αν. 10. (Scalar addition is distributive:) (a + B)u = au + Bu. 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 (f+9)(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 f given by f(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 x. The additive inverse of f is the function g(x)= for all x. (Scalar multiplication distributes over vector addition:) If c is any real number and f and g are two vectors in V, then c(f+9)(x) = c(f(x) + g(x)) =
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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Transcribed Image Text: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, w in V and for all
scalars a, ẞ in R.
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(u) = (aß)u.
9. (Scalar multiplication distributes over vector addition:) a(u+v)
= au + av.
10. (Scalar addition is distributive:) (a+8)u au + Bu.
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 +9)(±) = ƒ(±) + 9(2) for all real numbers
z. For any real number e and any function ƒ in V, define scalar multiplication of by
(ef)(a) cf(x) for all real numbers z.
=
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)+9(x) =
for all real numbers z since adding the real numbers f(z) and
9(a) is a commutative operation.
(A zero vector exists:) The zero vector in V is the function f given by f(x)
for all x.
(Additive inverses exist:) The additive inverse of the function f in V is a function g
that satisfies f(x)+9(x) = 0 for all real numbers z. The additive inverse off is the
function g(x) =
for all x.
(Scalar multiplication distributes over vector addition:) Ife is any real number and f
and g are two vectors in V, then e(f+9)(x) = c(f(x)+g(x))
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