/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 13 Let \(\mathbb{R}\) denote the se... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 13

Let \(\mathbb{R}\) denote the set of real numbers. Define scalar multiplication by \(\alpha x=\alpha \cdot x \quad\) (the usual multiplication of real numbers) and define addition, denoted \(\oplus,\) by \(x \oplus y=\max (x, y) \quad\) (the maximum of the two numbers Is \(R\) a vector space with these operations? Prove your answer.

Short Answer

Expert verified
\(\mathbb{R}\) is not a vector space with the given operations because there is no additive identity element.

Step by step solution

01

Verify associativity of addition

To verify associativity, we need to show that for any \(x, y, z \in \mathbb{R}\), we have \((x \oplus y) \oplus z = x \oplus (y \oplus z)\). Let's consider \(x, y, z \in \mathbb{R}\). Then: \((x \oplus y) \oplus z = \max(x, y) \oplus z = \max(\max(x, y), z)\). \(x \oplus (y \oplus z) = x \oplus \max(y, z) = \max(x, \max(y, z))\). Since the maximum operation is associative, we have \(\max(\max(x, y), z) = \max(x, \max(y, z))\). Therefore, the associativity of addition holds.
02

Verify commutativity of addition

To verify commutativity, we need to show that for any \(x, y \in \mathbb{R}\), we have \(x \oplus y = y \oplus x\). Let's consider \(x, y \in \mathbb{R}\). Then: \(x \oplus y = \max(x, y)\) \(y \oplus x = \max(y, x)\) Since the maximum operation is commutative, we have \(\max(x, y) = \max(y, x)\). Therefore, the commutativity of addition holds.
03

Find additive identity element

It turns out that there is no additive identity element in this case, since we need an element \(0 \in \mathbb{R}\) such that for any \(x \in \mathbb{R}\), \(x \oplus 0 = x\). However, consider any \(x > 0\), we have \(x \oplus 0 = \max(x, 0) = x > 0\). Thus, there is no element 0 for the given addition operation that behaves as an additive identity element. Since the existence of an additive identity element is not satisfied, we don't need to further verify other axioms. Thus, \(\mathbb{R}\) is not a vector space with these operations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scalar Multiplication
Scalar multiplication in the context of vector spaces involves taking a scalar and multiplying it by a vector. In this specific exercise, scalar multiplication is defined as the usual multiplication of real numbers. Thus, for any real numbers \(\alpha\) and \(x\), scalar multiplication is carried out by \(\alpha x = \alpha \cdot x\).
  • This operation is straightforward, simply performed by multiplying the scalar and the vector.
  • Scalar multiplication must satisfy the following properties, akin to classical multiplication:
  • Distributive: \(\alpha(x + y) = \alpha x + \alpha y\)
  • Associative: \((\alpha \beta) x = \alpha (\beta x)\)
  • Identity: \(1 \cdot x = x\)
Understanding these rules helps ensure the operation adheres to the standards of vector space operations.
Associativity
Associativity is a fundamental property of addition in any mathematical structure, including vector spaces. It requires the grouping of numbers in an addition operation not to affect the outcome. Specifically, for real numbers \(x, y, z\), associativity is shown through the equality \((x \oplus y) \oplus z = x \oplus (y \oplus z)\).
  • In this exercise, addition is defined as the maximum of two numbers.
  • The associativity is confirmed due to the property: \(\max(\max(x, y), z) = \max(x, \max(y, z))\).
  • Regardless of how the grouping is done, the maximum remains consistent.
This property is essential in demonstrating that a set with these operations could be considered a vector space.
Commutativity
Commutativity refers to the order of operations not affecting the result. For addition, this property means \(x \oplus y = y \oplus x\), which must hold for any pair of numbers in the space. In this specific operation, addition is defined by the maximum function.
  • Simply put, \(\max(x, y) = \max(y, x)\).
  • The maximum function is inherently commutative since the larger number remains the same regardless of their order.
  • This confirms the commutative property of addition here.
Commutativity is vital because it simplifies the structure and calculations within vector spaces.
Additive Identity
An additive identity is a special element within a set that, when added to any element of the set, results in that same element. In classical real numbers, zero is the additive identity because \(x + 0 = x\) for any real number \(x\).
  • The exercise attempts to find an additive identity using the maximum function as the addition operator.
  • It requires finding an element that satisfies \(x \oplus 0 = x\).
  • As shown, no element in the real numbers fulfills this property because \(x \oplus 0 = \max(x, 0)\) does not equal \(x\) when \(x > 0\).
Without meeting the additive identity requirement, the set fails to qualify as a vector space under these operations.
Real Numbers
Real numbers, denoted by \(\mathbb{R}\), form the basis for this exercise. They are a complete ordered field, meaning they can represent measurements with any level of precision without any gaps. Real numbers are used in this situation to define scalar multiplication and addition.
  • The real numbers include both rational numbers (fractions) and irrational numbers (such as \(\pi\) or \(\sqrt{2}\)).
  • They extend infinitely in both positive and negative directions and include zero.
  • The beauty of real numbers is their flexibility in mathematical operations while following a logical structure.
In this vector space examination, understanding real numbers is key, as they form the field over which vectors are defined and operated on.

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Most popular questions from this chapter

Given $$\mathbf{v}_{1}=\left(\begin{array}{l} 2 \\ 6 \end{array}\right), \quad \mathbf{v}_{2}=\left(\begin{array}{l} 1 \\ 4 \end{array}\right), \quad S=\left(\begin{array}{ll} 4 & 1 \\ 2 & 1 \end{array}\right)$$ find vectors \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\) so that \(S\) will be the transition matrix from \(\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\\}\) to \(\left\\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\\}\)

Let \(\mathbf{x}, \mathbf{y},\) and \(\mathbf{z}\) be vectors in a vector space \(V .\) Prove that if \\[ \mathbf{x}+\mathbf{y}=\mathbf{x}+\mathbf{z} \\] then \(\mathbf{y}=\mathbf{z}\)

Let \(\left\\{\mathbf{x}_{1}, \mathbf{x}_{2}, \ldots, \mathbf{x}_{k}\right\\}\) be a spanning set for a vector space \(V\) (a) If we add another vector, \(\mathbf{x}_{k+1},\) to the set, will we still have a spanning set? Explain. (b) If we delete one of the vectors, say \(\mathbf{x}_{k},\) from the set, will we still have a spanning set? Explain.

Let \(S\) be the vector space of infinite sequences defined in Exercise 15 of Section \(1 .\) Let \(S_{0}\) be the set of \(\left\\{a_{n}\right\\}\) with the property that \(a_{n} \rightarrow 0\) as \(n \rightarrow \infty\) Show that \(S_{0}\) is a subspace of \(S\).

Prove that any nonempty subset of a linearly independent set of vectors \(\left\\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{n}\right\\}\) is also linearly independent
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