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)) =
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.4: The Dot Product
Problem 45E
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