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Chapter 3: Problem 5

Show that \(C[a, b],\) together with the usual scalar multiplication and addition of functions, satisfies the eight axioms of a vector space.

Short Answer

Expert verified
To show that \(C[a, b]\) is a vector space, we verify that it satisfies all eight axioms of a vector space. First, we show that addition of functions in \(C[a, b]\) is commutative and associative. Next, we establish the zero function as the identity element for addition and the existence of additive inverses. Similarly, we examine the properties of scalar multiplication such as compatibility with field multiplication, existence of identity elements, and distributivity with respect to both vector and scalar addition. After confirming all these axioms hold true, we conclude that \(C[a, b]\) is indeed a vector space.

Step by step solution

01

Axiom 1: Commutativity of Addition

Let \(f, g \in C[a, b]\), then for all x in \([a, b]\), \((f + g)(x) = f(x) + g(x)\) and \((g + f)(x) = g(x) + f(x)\). Since \(f(x) + g(x) = g(x) + f(x)\), we have \((f + g)(x) = (g + f)(x)\). This shows that the addition of functions is commutative.
02

Axiom 2: Associativity of Addition

Let \(f, g, h \in C[a, b]\), then for all x in \([a, b]\), \(((f + g) + h)(x) = (f(x) + g(x)) + h(x)\) and \((f + (g + h))(x) = f(x) + (g(x) + h(x))\). Since \((f(x) + g(x)) + h(x) = f(x) + (g(x) + h(x))\), we have \(((f + g) + h)(x) = (f + (g + h))(x)\). This shows that the addition of functions is associative.
03

Axiom 3: Identity element of Addition

Consider the zero function, denoted by \(z(x) = 0\). For any \(f \in C[a, b]\), we have \((f + z)(x) = f(x) + z(x) = f(x) + 0 = f(x)\), and \((z + f)(x) = z(x) + f(x) = 0 + f(x) = f(x)\). Thus, the zero function acts as the identity element for function addition.
04

Axiom 4: Inverse elements of Addition

For any \(f \in C[a, b]\), consider the function \((-f)(x) = -f(x)\). We have \((f + (-f))(x) = f(x) + (-f(x)) = 0\), and \(((-f) + f)(x) = (-f(x)) + f(x) = 0\). So, the function \(-f\) functions as the inverse element of addition for \(f\).
05

Axiom 5: Compatibility of Scalar multiplication with Field multiplication

Let \(f \in C[a, b]\) and \(\alpha, \beta \in \mathbb{R}\). Then, for all x in \([a, b]\), \(((\alpha\beta) f)(x) = (\alpha\beta)f(x) = \alpha(\beta f(x)) = (\alpha (\beta f))(x)\), showing that scalar multiplication is compatible with field multiplication.
06

Axiom 6: Identity element of Scalar multiplication

Let \(f \in C[a, b]\). If we choose any scalar \(\alpha = 1\), then for all x in \([a, b]\), \((1 \cdot f)(x) = 1f(x) = f(x)\). Thus, the number 1 is the identity element for scalar multiplication.
07

Axiom 7: Distributivity of Scalar multiplication with respect to vector addition

Let \(f, g \in C[a, b]\) and \(\alpha \in \mathbb{R}\). Then, for all x in \([a, b]\), \((\alpha (f + g))(x) = \alpha(f(x) + g(x)) = \alpha f(x) + \alpha g(x) = (\alpha f + \alpha g)(x)\). This shows that scalar multiplication is distributive with respect to vector addition.
08

Axiom 8: Distributivity of Scalar multiplication with respect to scalar addition

Let \(f \in C[a, b]\) and \(\alpha, \beta \in \mathbb{R}\). Then, for all x in \([a, b]\), \(((\alpha + \beta) f)(x) = (\alpha + \beta)f(x) = \alpha f(x) + \beta f(x) = (\alpha f + \beta f)(x)\). This shows that scalar multiplication is distributive with respect to scalar addition. Since all eight axioms are satisfied, we can conclude that \(C[a, b]\), together with the usual scalar multiplication and addition of functions, forms a vector space.

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