Exploring Vector Space Over Reals

In summary, Vector space over reals is a mathematical concept that involves a set of vectors that can be manipulated using real numbers. It has properties such as closure, associativity, and commutativity, and has applications in various fields like physics, engineering, and computer science. It is different from vector space over complex numbers in terms of the scalars used. Some examples of vector space over reals include 2D and 3D vectors, polynomials, and continuous functions.
  • #1
Punkyc7
420
0
vector space?

Let v denote the set of order pairs of real numbers. If(a1,a2) and (b1,b2) are elements of V and c is an element of the reals, define (a1,a2)+(b1,b2)=(a1+b1,a2b2) and
c(a1,a2)=(ca1,a2)


is v a vector space over reals with these operations?

im thinking its not because the c only goes to a1

or because if a2=0 then there's no element in b2 that makes 1
 
Last edited:
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  • #2


satrt by listing the vector space axioms and see if the operations satisfy them
 

Related to Exploring Vector Space Over Reals

What is vector space over reals?

Vector space over reals is a mathematical concept that refers to a set of vectors that can be added and multiplied by real numbers. It is a fundamental concept in linear algebra and has applications in physics, engineering, and computer science.

What are the basic properties of vector space over reals?

The basic properties of vector space over reals include closure, associativity, commutativity, distributivity, and the existence of an identity element. These properties ensure that the set of vectors is well-defined and can be manipulated using standard mathematical operations.

How is vector space over reals different from vector space over complex numbers?

The main difference between vector space over reals and vector space over complex numbers is the type of scalars used. In vector space over reals, the scalars are real numbers, while in vector space over complex numbers, the scalars are complex numbers. This leads to different properties and applications of the two types of vector spaces.

What are the applications of exploring vector space over reals?

Exploring vector space over reals has many applications in various fields, such as physics, engineering, computer graphics, and data analysis. In physics, it is used to describe the movement of objects in physical space. In engineering, it is used for control systems and optimization problems. In computer graphics, it is used for 3D modeling and animation. In data analysis, it is used for data visualization and machine learning algorithms.

What are some examples of vector space over reals?

Some examples of vector space over reals include the set of all 2D and 3D vectors with real number components, the set of all polynomials with real coefficients, and the set of all continuous functions on a real interval. These are just a few examples, as there are infinitely many vector spaces over reals that can be constructed.

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