The vector space status of all nonconstant functions

In summary, Dick tries to solve a homework problem, but does not understand how the order of operations works and ends up getting wrong answers. He asks for help from the community, and is assisted by a fellow redditor. Together, they figure out that the problem is not solvable and that the set of all nonconstant functions is not a subset of the set of all functions.
  • #1
MoreDrinks
45
0

Homework Statement


Let V be the set of all nonconstant functions with operations of pointwise addition and scalar multiplication, having the real numbers as their domain. Is V a vectorspace?


Homework Equations


None.


The Attempt at a Solution


My guess is, no. For example
F(x) = x2
Among the axioms for vector space, for an arbitrary element of a vector space u there must be a -u. We can put any number in x and it will end up positive. We can multiply it by any negative scalar, but then squaring it will make it positive.

Am I right? Am I wrong? Am I right for the wrong reasons?

If I am right, there's another disturbing question that comes up: the explanatory portions of the chapter I'm working on state that "The set of all functions having the real numbers as their domain, with operations of pointwise addition and scalar multiplication, is a vector space." Well, isn't the set of all nonconstant functions a subset of that set?
 
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  • #2
MoreDrinks said:

Homework Statement


Let V be the set of all nonconstant functions with operations of pointwise addition and scalar multiplication, having the real numbers as their domain. Is V a vectorspace?


Homework Equations


None.


The Attempt at a Solution


My guess is, no. For example
F(x) = x2
Among the axioms for vector space, for an arbitrary element of a vector space u there must be a -u. We can put any number in x and it will end up positive. We can multiply it by any negative scalar, but then squaring it will make it positive.

Am I right? Am I wrong? Am I right for the wrong reasons?

If I am right, there's another disturbing question that comes up: the explanatory portions of the chapter I'm working on state that "The set of all functions having the real numbers as their domain, with operations of pointwise addition and scalar multiplication, is a vector space." Well, isn't the set of all nonconstant functions a subset of that set?


Right for the wrong reason. Nonconstant functions are a subset (not a subspace) of all functions. But it's not closed. Find two nonconstant functions that you can add to get a function that is non-nonconstant, i.e. constant.
 
  • #3
Dick said:
Right for the wrong reason. Nonconstant functions are a subset (not a subspace) of all functions. But it's not closed. Find two nonconstant functions that you can add to get a function that is non-nonconstant, i.e. constant.

Oh, okay. So, for example

F(x)= x+1
G(x)= -x +1

Add them together and get a constant, 1. Is that correct?

If you have a moment, what was wrong with my original reasoning with F(x)=x^2
 
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  • #4
MoreDrinks said:
Oh, okay. So, for example

F(x)= x+1
G(x)= -x +1

Add them together and get a constant, 1. Is that correct?

If you have a moment, what was wrong with my original reasoning with F(x)=x^2

Yes, that's correct. Except if add them you get 2. As for your first reasoning I simply don't understand it. kx^2 isn't the same thing as (kx)^2.
 
  • #5
I agree with Dick that you are "right for the wrong reason". The lack of closure is a killer.

Another killer: What's your zero vector, the additive identity? It can only be f(x)=0, but that's a constant function, so it's not in your set.
 
  • #6
D H said:
I agree with Dick that you are "right for the wrong reason". The lack of closure is a killer.

Another killer: What's your zero vector, the additive identity? It can only be f(x)=0, but that's a constant function, so it's not in your set.

You are correct sir! Thanks.
 
  • #7
Dick said:
Yes, that's correct. Except if add them you get 2. As for your first reasoning I simply don't understand it. kx^2 isn't the same thing as (kx)^2.

Whoops, yeah, two. Thank you for your help.

Oh, am I just wrong on the order of operations - the exponential would come first in kx^2, then multiplication by the scalar, which could leave us with negatives?
 
Last edited:
  • #8
MoreDrinks said:
Whoops, yeah, two. Thank you for your help.

Oh, am I just wrong on the order of operations - the exponential would come first in kx^2, then multiplication by the scalar, which could leave us with negatives?

Yes. Exactly.
 
  • #9
Dick said:
Yes. Exactly.

Thanks for all the help. I usually end up coming here when I'm exhausted and frustrated, and then we see sloppy mistakes on my part. You're a tremendous help.
 

FAQ: The vector space status of all nonconstant functions

What is a vector space?

A vector space is a mathematical structure that consists of a set of objects called vectors, along with operations such as addition and scalar multiplication. These operations must satisfy certain properties, such as closure and associativity, in order for the set to be considered a vector space.

2. How are nonconstant functions related to vector spaces?

Nonconstant functions can be thought of as vectors in a vector space, where the domain of the function serves as the basis for the space. This means that the functions can be added, scaled, and manipulated using vector operations.

3. What is the significance of the vector space status of nonconstant functions?

The vector space status of nonconstant functions allows us to use techniques and concepts from linear algebra to study and understand these functions. It also provides a framework for solving problems involving nonconstant functions in a more efficient and systematic manner.

4. Are there any limitations to considering nonconstant functions as vectors in a vector space?

Yes, there are limitations. For example, the vector space of nonconstant functions does not include functions with discontinuities or infinite values. Additionally, not all operations that can be performed on vectors in a traditional vector space may be applicable to nonconstant functions.

5. How does the vector space status of nonconstant functions relate to other areas of mathematics?

The concept of vector spaces is a fundamental one in mathematics, and the study of nonconstant functions as vectors in a vector space has connections to other areas such as functional analysis, differential equations, and optimization. It also has applications in fields such as physics, engineering, and computer science.

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