Is R a Vector Space with Defined Operations? | Homework Statement

In summary, the set of real numbers with scalar multiplication defined as regular scalar multiplication and vector addition defined as the maximum value of two numbers is not a vector space due to the lack of a zero vector.
  • #1
amolv06
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Homework Statement



Let R denote the set of real numbers. Define scalar multiplication by [tex]\alpha x = \alpha x[/tex] which is simply regular scalar multiplication, and vector addition is defined as [tex]x \oplus y = max(x,y)[/tex]. Is R a vector space with these operations?

Homework Equations



Some given above.

The Attempt at a Solution



There seems to be no zero vector to this equation as for any number k there exists another number k-1, so there is no single 0 vector for a vector space with the operations defined above. Is this reasoning correct?
 
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  • #2
Yes, that's true. So what is your answer to the question?
 
  • #3
Then it is not a proper vector space!

Thanks a lot.
 

1. What is a vector space?

A vector space is a mathematical structure that consists of a set of vectors and operations that define how those vectors can be combined and manipulated. In order for a set to be considered a vector space, it must satisfy certain axioms, including closure under addition and scalar multiplication, and the existence of a zero vector and additive inverses.

2. How do you define operations in a vector space?

Operations in a vector space are defined based on the properties of the vectors in the set. Addition is defined as the combining of two vectors to create a new vector, while scalar multiplication is defined as multiplying a vector by a scalar (a real or complex number). These operations must satisfy certain properties, such as commutativity and associativity, in order for the set to be considered a vector space.

3. What is the role of a zero vector in a vector space?

The zero vector is an important component of a vector space because it serves as the additive identity element. This means that when the zero vector is added to any other vector, the result is the original vector. It also ensures that every vector space has at least one vector.

4. What are some examples of vector spaces?

Examples of vector spaces include the set of real numbers, the set of complex numbers, the set of all n-dimensional vectors, and the set of polynomials with real or complex coefficients. Vector spaces can also be defined in more abstract contexts, such as the set of continuous functions or the set of all square matrices.

5. Is R (the set of real numbers) a vector space with defined operations?

Yes, R is a vector space with defined operations. The set of real numbers satisfies all of the axioms of a vector space, including closure under addition and scalar multiplication, and the existence of a zero vector and additive inverses. This means that R can be used as a vector space in mathematical calculations and proofs.

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