# Linear algebra: vector spaces

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• #1
phys2
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## Homework Statement

Hi, I am really having trouble with questions regarding proving whether a given set is a vector space or not.

So one of the questions is [ x ε R2|x12=x23 ]

So I have to prove whether the following set is a vector space

## The Attempt at a Solution

Having a peek at the textbook, it tells me that there are ten axioms that satisfy whether a set belongs to a vector space.

What I did was go x12 - x23 = 0

u+ v = v + u -For this one, I did the following: I let x1 and x2 have a value each and named it a vector u, then I let x1 and x2 have some other values and named it a vector v.

So x1 = 1 and x2 = 2 and u = 12 - 23
And x1 = 4 and x2 = 5 and v = 42 - 53

u + v = v + u should work out and I used the same method for the rest of the axioms....I need to know two things

(a) Is the method I am doing correct? If not, could you point me in the right direction?
(b) How do I go about checking the first axiom for the above set u + v lies in the set. If I do the same old substitution, all i get is a number, whatever it is!

Thanks

## The Attempt at a Solution

• #2
Homework Helper
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Many of the axioms will be satisfied automatically because this set (let's give it a name: $S$) is a subset of a vector space.

For example, if $u$ and $v$ are in $S$, then they are also in $\mathbb{R}^2$, and since $\mathbb{R}^2$ is a vector space, addition is commutative. Therefore $u + v = v + u$.

The main things to check are whether $S$ is closed under vector addition and scalar multiplication. The fact that $S$ is defined in terms of squares and cubes of the vector coordinates is a pretty good indication that $S$ is not a vector space. This is because squaring and cubing are both nonlinear operations, and vector spaces are all about linearity.

Try picking a specific element of $S$, for example $v = (1,1)$. Is there a scalar multiple of $v$ that is not in $S$? Is $v + v$ in $S$?

• #3
phys2
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Yea, that is really the part (b) of the problem that I faced. I don't think I can take v = (1,1) though since both numbers have to be different x12 - x23 Anyway, even if I take two different numbers, what I get is another number. How do I know whether that number is part of set S or not?

For instance, if I take v = (1,2), I will get 1-8 = -7. How do I know -7 is in S?

I did check all the rules using the method that you have mentioned, taking numbers and figuring out whether the axioms are satisfied...but I wasn't sure whether that was a reasonable method...but since you argue in the same way, I guess it is.

• #4
Homework Helper
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Yea, that is really the part (b) of the problem that I faced. I don't think I can take v = (1,1) though since both numbers have to be different x12 - x23 Anyway, even if I take two different numbers, what I get is another number. How do I know whether that number is part of set S or not?

For instance, if I take v = (1,2), I will get 1-8 = -7. How do I know -7 is in S?
I think you're misunderstanding the condition $x_1^2 = x_2^3$. The vector $v$ is in $S$ if and only if its components satisfy $x_1^2 = x_2^3$. In your example, $x_1 = 1$ and $x_2 = 2$, so $x_1^2 = 1$ and $x_2^3 = 8$. Since $1 \neq 8$, this means that $v$ is not in $S$.

Now consider my example again: $v = (1,1)$. In this case, $x_1 = 1$ and $x_2 = 1$, so $x_1^2 = 1$ and $x_2^3 = 1$. Therefore $x_1^2 = x_2^3$, so $v \in S$.

So now that we have established that $v = (1,1)$ is in $S$, you can easily use this to show that $S$ is not closed under addition and scalar multiplication, so $S$ is not a vector space. It doesn't matter that it satisfies the other axioms. If it isn't closed under addition and scalar multiplication, it cannot be a vector space.

To show that $S$ is not closed under addition and scalar multiplication, consider the vector $v + v = (2,2)$. If $S$ is a vector space, then $(2,2)$ must be in $S$. Is it?

Last edited:
• #5
phys2
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To show that S is not closed under addition and scalar multiplication, consider the vector v+v=(2,2). If S is a vector space, then (2,2) must be in S. Is it?

(2,2) is not in S because 22 = 4 and 23 = 8 and 4≠8.

I am still confused about showing that S is not closed under addition and scalar multiplication.

With addition, I know that the axiom is u + v = v + u

So if I take v = (2,2), u = (1,2), do I just do it the normal vector addition way; adding the x1s together and the x2s together?

So v+ u = [ (22 + 12) , ( 23 + 23) ] = (5, 16)

u + v= [ (12 + 22) , (23 + 23) ] = (5,16)

Seems like it is closed under vector addition or I am probably doing something wrong?

• #6
Homework Helper
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So if I take v = (2,2), u = (1,2), do I just do it the normal vector addition way; adding the x1s together and the x2s together?

So v+ u = [ (22 + 12) , ( 23 + 23) ] = (5, 16)
Addition is defined in the usual way: $(2,2) + (1,2) = (2+1, 2+2) = (3,4)$. The $x_1^2 = x_2^3$ relation is just a constraint which tells you which vectors are in $S$. It has nothing to do with how you add vectors or multiply by scalars.

• #7
phys2
22
0
The x21=x32 relation is just a constraint which tells you which vectors are in S. It has nothing to do with how you add vectors or multiply by scalars.

Oh, I think I get it. What we have done so far is find vectors that belong or do not belong to S. Example: v = (1,2) does not belong to S and v= (1,1) belongs to S.

But v = (1,1), even if it belongs to S, that does not necessarily make it a vector space, I need to do do the addition and scalar multiplication to figure it out. Is my understanding correct so far?

If S is a vector space, then (2,2) must be in S.

v = (2,2) is not in S so S is not a vector space? What I know about vector spaces is that, take R2 for an example. R2 is a vector space because you can take any linear combinations of vectors (x1, x2) and they all lie in R2.

• #8
Homework Helper
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Oh, I think I get it. What we have done so far is find vectors that belong or do not belong to S. Example: v = (1,2) does not belong to S and v= (1,1) belongs to S.

But v = (1,1), even if it belongs to S, that does not necessarily make it a vector space, I need to do do the addition and scalar multiplication to figure it out. Is my understanding correct so far?
Yes, that's right.

v = (2,2) is not in S so S is not a vector space? What I know about vector spaces is that, take R2 for an example. R2 is a vector space because you can take any linear combinations of vectors (x1, x2) and they all lie in R2.
Right. Now, we already established that v = (1,1) is in S. Now let's take a linear combination of elements of S: (1,1) + (1,1) = (2,2) is a linear combination, namely 1v + 1v. If S is a vector space, then S must contain all linear combinations of its elements. Therefore if (2,2) is not in S, then S cannot be a vector space.

• #9
phys2
22
0
Yes, I finally get it now. Using the same method, I should be able to get through all the problems.

Thanks a lot for your help!

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