Is V a Vector Space? A Quick Question on Axiom 8 Explained

In summary, this set is not a vector space because it is not closed under scalar multiplication and axiom 8 does not hold.
  • #1
kwal0203
69
0

Homework Statement



Determine whether this set equipped with the given operations is a vector space. For those that are not vector spaces identify the axiom that fails.

Set = V = all pairs of real real numbers of the form (x,y) where x>=0, with the standard operations on R^2.


The Attempt at a Solution



This set is not a vector space because it is not closed under scalar multiplication I.e. -1*(1,1)=(-1,-1) which is not in V as x<0 and because there is not always a vector in V such that u+(-u)=(-u)+u=0 I.e. when u=(1,1) then -u=(-1,-1) which is not in V as again x<0.

My question is why does axiom 8 hold which states:

(K+m)u=ku+km

I.e. if k=-1, m=-1, u=(1,1) ----> (-1+-1)u=(-1,-1)+(-1,-1)=(-2,-2) which is not in V as x<0.

Does axiom 8 not require the solution to be in the set V?
 
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  • #2
You showed that the space is not closed under scalar multiplication, so it's not a vector space. But also, since it's not closed under scalar multiplication, all the axioms which use scalar multiplication make no sense. It makes no sense to asl ##(\alpha + \beta) v = \alpha v + \beta v##, since scalar multiplication is not well-defined.

so I wouldn't say that Axiom 8 holds in this case. I would rather say that it makes no sense.
 
  • #3
micromass said:
You showed that the space is not closed under scalar multiplication, so it's not a vector space. But also, since it's not closed under scalar multiplication, all the axioms which use scalar multiplication make no sense. It makes no sense to asl ##(\alpha + \beta) v = \alpha v + \beta v##, since scalar multiplication is not well-defined.

so I wouldn't say that Axiom 8 holds in this case. I would rather say that it makes no sense.

Oh I see, but it could be the case that a set is closed under scalar multiplication but one of the axioms that depend on that, such as axiom 8, do not hold. Just not vice versa.

Thanks!
 

1. What is a vector space?

A vector space is a mathematical concept that refers to a set of vectors that can be added and scaled by real numbers. It is an important concept in linear algebra and is used to represent physical quantities such as force, velocity, and acceleration.

2. How do you determine if a set of vectors forms a vector space?

To determine if a set of vectors forms a vector space, you need to check if it satisfies the 10 properties of a vector space. These include closure under addition and scalar multiplication, existence of a zero vector, and existence of additive and multiplicative inverses.

3. Can you have a vector space in more than three dimensions?

Yes, a vector space can exist in any number of dimensions, including more than three. In fact, vector spaces are often used in higher dimensions, such as in computer graphics and quantum mechanics.

4. What is the difference between a vector space and a vector?

A vector is an element within a vector space, while a vector space is a collection of vectors. Think of a vector space as a container that holds multiple vectors, which are individual objects with magnitude and direction.

5. How are vector spaces used in scientific research?

Vector spaces have numerous applications in scientific research, particularly in fields such as physics, engineering, and computer science. They are used to model physical systems, analyze data, and solve complex problems. For example, vector spaces are used in quantum mechanics to represent the state of a particle, and in machine learning to classify data points.

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