Is V Also a Vector Space Over the Real Numbers?

In summary, the conversation discusses whether V, a set of vectors with complex number elements, is a vector space over the field of real numbers with the operations of coordinatewise addition and multiplication. The answer is yes, as long as the set of complex numbers satisfies the definition of a vector space over a field, which it does. The confusion may have come from misunderstanding the definition and the fact that the field only restricts the scalars we can multiply by, not the possible results.
  • #1
loli12
Let V = {(a1, a2, ..., an): ai in C for i = 1, 2, ... n}; (C=complex numbers) ; so, V is a vector space over C. Is V a vector space over the field of real numbers with the operaions of coordinatewise addition and multiplication?

I thought the answer to this question is No since after we perform the operations on any two of the elements in C, we are getting some other complex numbers which is not an element in R. But the book says Yes to the question. Can anyone please tell me what's wrong with my concept and what is the correct way to brainstorm this question?

Thanks!
 
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  • #2
does anyone has any idea to the problem? or is there any other missing information that I have to provide? Please Please help!
 
  • #3
Look at the DEFINITION of "vector space over field F".

A vector space over a field is:

A set of objects with addition and scalar multiplication defined such that
1) For all x, y in the set, x+ y is also in the set and all the "group" properties hold.
Okay, if our set is the set of complex numbers the sum is a complex number and it is easy to show that the set of complex numbers, with usual addition, is a group.
In fact, as far as the properties of addition are concerned, the field is irrelevant!

2) For all x in the set, a in the field, ax is in the set, a(x+y)= ax+ay, (a+b)x=ax+ bx (the "distributive laws).
Sure: if x is a complex number, a is a real number, ax is a complex number. The distributive laws hold for all complex numbers and the real numbers are a subset of complex numbers so it doesn't matter if we restrict a to be real!

"I thought the answer to this question is No since after we perform the operations on any two of the elements in C, we are getting some other complex numbers which is not an element in R."

The vectors are in C, not R. It's only the "scalars" we multiply by that have to be in R. Making R the field means we restrict what we can multiply by, not the possible result.
 

FAQ: Is V Also a Vector Space Over the Real Numbers?

What is a vector space over a field of R?

A vector space over a field of R is a mathematical structure that consists of a set of elements (vectors) that can be added together and multiplied by a scalar (a real number) to produce a new vector. The field of R refers to the set of real numbers, and a vector space over this field is denoted as V(R).

What are the axioms of a vector space over a field of R?

The axioms of a vector space over a field of R are a set of rules that must be satisfied in order for a set to be considered a vector space. These axioms include closure under addition and scalar multiplication, commutativity and associativity of addition, existence of a zero vector, existence of an additive inverse, distributivity of scalar multiplication over vector addition, and compatibility of scalar multiplication.

How is a vector space over a field of R different from a vector space over a different field?

The main difference between a vector space over a field of R and a vector space over a different field is the set of scalars that can be used for scalar multiplication. In a vector space over a field of R, the scalars are limited to real numbers, while in a vector space over a different field, the scalars can be any elements from that field. This leads to different properties and characteristics for each type of vector space.

What are some examples of vector spaces over a field of R?

Some examples of vector spaces over a field of R include Euclidean space (n-dimensional space), the set of polynomials with real coefficients, and the set of real-valued functions defined on a given interval. Other examples include the set of matrices with real entries and the set of solutions to a homogeneous linear differential equation.

How are vector spaces over a field of R used in real-world applications?

Vector spaces over a field of R have a wide range of applications in fields such as physics, engineering, and computer science. They are used to model physical quantities such as force and velocity, analyze linear systems and equations, and represent data in computer programming. They are also fundamental in linear algebra, a branch of mathematics that has numerous applications in various fields.

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