Is V a Vector Space with These Operations?

Note that the problem as written is incomplete; we need to know precisely how the addition is defined. I would guess it's the obvious way, but who knows.)In summary, the set V is defined as the set of all functions from R to R with pointwise addition and a unique scalar multiplication operation. However, it does not satisfy the distributivity of field addition over scalar multiplication axiom, as shown by the counter-example of f(x) = x^2. Therefore, V is not a vector space.
  • #1
skoomafiend
33
0

Homework Statement



V is the set of functions R -> R; pointwise addition and (a.f)(x) = f(ax) for all x.

is V a vector space given the operations?

Homework Equations



nil.

The Attempt at a Solution



i think it is not closed under multiplication.
if r is an element of R, then
r*a(x) . r*f(x) = (r^2)*(a.f)(x)
which is not equal to
r*f(ax)

im not really sure if i even have the correct approach.
any help would be greatly appreciated.

thanks!
 
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  • #2
skoomafiend said:

Homework Statement



V is the set of functions R -> R; pointwise addition and (a.f)(x) = f(ax) for all x.

is V a vector space given the operations?


Homework Equations



nil.

There are lots of relevant equations -- they comprise the definition of a vector space.

The Attempt at a Solution



i think it is not closed under multiplication.
if r is an element of R, then
r*a(x) . r*f(x) = (r^2)*(a.f)(x)
which is not equal to
r*f(ax)

im not really sure if i even have the correct approach.
any help would be greatly appreciated.

thanks!

I don't follow your argument. There is no "multiplication" of vectors in the definition of a vector space, only addition. All you need to do is pick one of the properties of a vector space that doesn't work and give a counter-example. Which property are you working with above? It might be useful to list them.
 
  • #3
suppose that f(x) = x2.

is it the case that ((a+b).f)(x) = (a.f)(x) + (b.f)(x)?

(this is the distributivity of field addition over scalar mutliplication axiom).
 
  • #4
skoomafiend said:

Homework Statement



V is the set of functions R -> R; pointwise addition and (a.f)(x) = f(ax) for all x.

Deveno said:
suppose that f(x) = x2.

is it the case that ((a+b).f)(x) = (a.f)(x) + (b.f)(x)?

(this is the distributivity of field addition over scalar mutliplication axiom).
If I'm understanding the problem correctly, f(x) = x2 is not a member of set V, since af(x) [itex]\neq[/itex] f(ax).
 
  • #5
Mark44 said:
If I'm understanding the problem correctly, f(x) = x2 is not a member of set V, since af(x) [itex]\neq[/itex] f(ax).

You aren't. The vector space is the set of [all] functions from R to R. It's just that scalar multiplication is defined in an unusual way.
 

FAQ: Is V a Vector Space with These Operations?

1. What is a vector space of functions?

A vector space of functions is a mathematical structure that consists of a set of functions and two operations, addition and scalar multiplication, that satisfy certain properties. These properties include closure under addition and scalar multiplication, associativity, commutativity, and the existence of an identity element and inverse element.

2. How are vector spaces of functions different from regular vector spaces?

The main difference between vector spaces of functions and regular vector spaces is the type of elements they contain. Regular vector spaces contain vectors, which are defined as a combination of numbers, while vector spaces of functions contain functions as their elements. Additionally, vector spaces of functions have operations defined on them, such as addition and scalar multiplication, that are specific to functions.

3. What are some common examples of vector spaces of functions?

Some common examples of vector spaces of functions include the space of all polynomials of a certain degree, the space of all continuous functions on a given interval, and the space of all differentiable functions on a given interval. These spaces are often used in mathematical analysis and are essential in many areas of mathematics and physics.

4. What is the dimension of a vector space of functions?

The dimension of a vector space of functions is the number of linearly independent functions that span the space. This is similar to the dimension of a regular vector space, which is the number of linearly independent vectors that span the space. However, in a vector space of functions, the basis elements are functions rather than vectors.

5. How are vector spaces of functions used in real-world applications?

Vector spaces of functions have many practical applications in fields such as engineering, physics, and computer science. For example, they are used in signal processing to analyze and manipulate signals, in control theory to model and control systems, and in computer graphics to represent and manipulate images and shapes. They are also heavily used in mathematical physics, where they provide a powerful framework for representing and solving physical problems.

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