How can I tell if this is a vector space?

[Antineutron]

1. Homework Statement
A set of objects is given, together with operations of addition and scalar multiplication. Determine which sets are vector spaces under the given operations. For those that are not vector spaces, list all axioms that fail to hold.

(x,y,z) + (x',y',z') = (x+x',y+y',z+z') and k(x,y,z) = (kx,y,z)


2. Homework Equations

1. If u and v are objects in V, then u + v is in V.

2. u + v = v + u

3. u + (v + w) = (u + v) + w

4. There is an object 0 in V, called a zero vector for V, such that 0 + u = u+ 0 = for all un in V

5. For each u in V, there is an object -u in V, called a negative of u, such that u + (-u) = (-u) + u = 0.

6. If k is any scalar and u is any object in V, then ku is in V.

7. k(u + v) =ku + kv

8. (k + m)u = ku + mu

9. k(mu) = (km)(u)

10. 1u=u

3. The Attempt at a Solution

I have no idea what they are asking for, the back of the books say it fails one axiom which is: (k+m)u = ku +mu Which is axiom 8 in my book.

 

[cristo]

Well, since you say there are ten axioms for a vector space, then you should check whether the set satisfies the axioms.

 

[Antineutron]

thats the part I am having trouble with. How do see that it fails the axiom that I have just stated? Can you show me?

 

[JasonRox]

Don't you have any examples in the textbook?

Do you understand what the axioms are?

List the axioms.

Note: By the way, the worse thing you can do is look up the answer before trying or even understanding the question.

 

[Antineutron]

I agree, but I'm not the kind of student that looks up answer unless I am desperate. I'm not a precalculus student you know. And this is not a simple calculation question. Its just a simple question that I cannot comprehend. Also, there is no examples that are anything like this question in my textbook. Thank you.

Here are the axioms:

1. If u and v are objects in V, then u + v is in V.

2. u + v = v + u

3. u + (v + w) = (u + v) + w

4. There is an object 0 in V, called a zero vector for V, such that 0 + u = u+ 0 = for all un in V

5. For each u in V, there is an object -u in V, called a negative of u, such that u + (-u) = (-u) + u = 0.

6. If k is any scalar and u is any object in V, then ku is in V.

7. k(u + v) =ku + kv

8. (k + m)u = ku + mu

9. k(mu) = (km)(u)

10. 1u=u

Thats all, did you ever had time when doing math homework, that you could not understand something, but you know its not so difficult. Just remember that time, and that's how I feel right now.

 

[morphism]

OK. So you have a set S of triples (x,y,z) (you should be clear about where x,y,z are coming from -- you didn't state this in your post; I'm going to assume that they're real numbers). Then two binary operations are being defined on S:
"addition": (x, y, z) + (x', y', z') = (x+x', y+y', z+z')
and "scalar multiplication": k*(x, y, z) = (kx, y, z)

Now if S is a vector space under these two operations, it has to satisfy all the axioms you listed. For example, let's try to see if it satisfies the first. Let u=(x,y,z) and v=(x',y',z'). We want to prove that u+v is in S. Since u+v=(x+x',y+y',z+z') by definition, and since x+x', y+y' and z+z' are all real numbers, then u+v is in S. So the first axiom holds.

Can you take it from here?

 

[Antineutron]

I understand that part, but can you tell me why axiom 8 does not hold? Thats where I am having trouble. How do I appy axiom 8 to test to see if the scalar multiplication that is stated does is not vector space according to that particular axiom?

 

[ZioX]

Argh. Please don't cross post!

You would want to find a counterexample, try playing around with it.

 

[Antineutron]

Look at axiom 8 and tell me in good mathematically rigorous explanation why that one does not apply to the scalar multiplication. Thats all.

 

[D H]

Expand each side of (k+m)*u = ku + mu to see whether this holds for any k, m, and u=(x,y,z). Look at the non-trivial case: none of k,m, x, y, or z are zero.

 

[ice109]

(k+m)u=?ku+mu
((k+m)x,y,z)=(kx,y,z)+(mx,y,z)
((k+m)x,y,z)=((k+m)x,2y,2z)

therefore fails

as opposed to vector space with scalar multiplication defined like this: k(x,y,z)=(kx,ky,kz)
(k+m)u=?ku+mu
((k+m)x,(k+m)y,(k+m)z)=(kx,ky,kz)+(mx,my,mz)
((k+m)x,(k+m)y,(k+m)z)=((k+m)x,(k+m)y,(k+m)z) =true

 

[D H]

How could it fail axiom 6? V is a vector space over a field F. This means if u=(x,y,z) is are in V, each of x, y, and z are in F. Multiplication by a scalar means a scalar in the underlying field F. Thus k, x, y, and z are each in F. The product kx is in F, so ku=(kx,y,z) is in V.

 

[ice109]

How could it fail axiom 6? V is a vector space over a field F. This means if u=(x,y,z) is are in V, each of x, y, and z are in F. Multiplication by a scalar means a scalar in the underlying field F. Thus k, x, y, and z are each in F. The product kx is in F, so ku=(kx,y,z) is in V.

i misread 6 not as closure under addition but something else

 

[cristo]

Look at axiom 8 and tell me in good mathematically rigorous explanation why that one does not apply to the scalar multiplication. Thats all.

I know ice109 has given you the answer, but you should not be demanding something like this. People are here helping you with your homework; you should not expect people to be doing it for you.

This is a standard type introductory question in vector spaces: one is given a set and required to check whether it satisfies the axioms of a vector space. You have listed the axioms, but then not made any attempt at checking them. I suggest in future that you make some effort before posting, and do not demand that someone do your work for you.

 

[mathwonk]

you are in over your head. where did this assignment come from in precalculus? try some more elementary and basic stuff like simple logic.

 

[D H]

Antineutron, we have rules against posting the same problem in multiple forums. You violated that rule with this thread and https://www.physicsforums.com/showthread.php?t=194564. Please don't do this again.

 

[Antineutron]

I'm sorry, I needed some help really bad, I will not double post again. I did not post to get an answer without trying to find the answer. I just had a hard time formulating a good solid proof to varify all the given axioms, axiom 8 was not apparent as the others when applying it to this problem, it is not that the axioms are difficult. I ask you one thing mathwork, if you think this is simple logic, did you know all the math before going to college? Is something simple logic, so we should all know everything. Are we to know everying because it is logic? What is the point of going to school and learning? What is your point of telling me is simple logic?

 

[matt grime]

Perhaps mathwonk's point was that the negation of 'for all' is 'there exists' and you thus need only to find one counter example, and has little to do with the theory of vector spaces at all. Pretty much any thing you try will yield a counter example to 8. Thus the inferrence is that you didn't multiply any notional vectors by any scalars. Not playing with the example to see what is going on is the worst thing you can do: answers do not often magically appear. In fact mathwonk is arguably saying the exact opposite of what you ascribe to him: it isn't about knowing all the answers but about experimenting with things to see what happens.

Multiplication by 2 is adding something to itself. Clearly those two descriptions, in this case, are incompatible.

 

[ice109]

you are in over your head. where did this assignment come from in precalculus? try some more elementary and basic stuff like simple logic.

he said he's not in precalc? besides what about linear algebra do you think is too abstruse for pre calc students? heck after taking lin alg i think this course could be taught immediately after algebra in middle/high school

 

[mathwonk]

after reading post 17, i am more sure than before you are over your head. why are you so indignant that i tell you the truth? get a book on logic and read some of it.

you apparently have no experience at all with axiom systems, or definitions, or counterexamples as matt suggests.

 

[Antineutron]

after reading post 17, i am more sure than before you are over your head. why are you so indignant that i tell you the truth? get a book on logic and read some of it.

you apparently have no experience at all with axiom systems, or definitions, or counterexamples as matt suggests.

Bingo, I don't, thank you for pointing that out. All the books you read made you who you are mathwonk, I'm sure you are very proud. Should I get a book on logic like you say so I can learn from books like you?

 

[D H]

Enough of the trash talk already! I do not want to see fisticuffs at the PF Saloon.

Mathwonk: Telling someone in public that they are in over their heads is an insult. Please desist.

Antineutron: Even though ice109 gave out the answer, please work through how axiom #8 fails in general here. Aside: Being in over your head is not necessarily a bad thing. It forces one to learn and achieve better than almost anything else.

Ice109: Doggone it, stop giving out answers. This was not your thread.

 

[Antineutron]

I went over the problems and tried to figure it out after looking at ice's example which was very helpful. I really appreciate the people that helped me figure out this new way of looking at mathematical concepts, that includes you DH. I feel gracious that I can come here for help and get help from people that can understand people's hardships and not judge people because we may not be so experienced. And I'm sure many people have been in that road before succeeding in what ever areas many of you professionals are in now.

 

[ice109]

Enough of the trash talk already! I do not want to see fisticuffs at the PF Saloon.

Mathwonk: Telling someone in public that they are in over their heads is an insult. Please desist.

Antineutron: Even though ice109 gave out the answer, please work through how axiom #8 fails in general here. Aside: Being in over your head is not necessarily a bad thing. It forces one to learn and achieve better than almost anything else.

Ice109: Doggone it, stop giving out answers. This was not your thread.

honestly he can't begin to do these types of problems if he has no idea what the solutions even look like.

 

[cristo]

he said he's not in precalc? besides what about linear algebra do you think is too abstruse for pre calc students? heck after taking lin alg i think this course could be taught immediately after algebra in middle/high school

But that's what everyone says after they've learned a course. The fact of the matter is that the majority of students in school firstly, do not like maths, and secondly, are not very good at maths. So attempting to teach linear algebra to them would just be a complete waste of time and effort.

 

[cristo]

honestly he can't begin to do these types of problems if he has no idea what the solutions even look like.

And there's even less chance of someone learning if you simply give them answers, is there?

 

[Antineutron]

I'm sorry guys, I didn't begin this thread to open up any philosophies the tutors in these forums have about how somthing should be learned. But, let me tell you that I like math. I started from precalc and went through calculus 1-3, then differential equations was even much easier because of my calc 2 background. I'm getting a good grade in this class, but why am I struggling with this concept? I wish I had the answers and correct my difficulty. I remember when taking calc for the first time, I had fears of limits, and only by looking at teacher's example could I even get used to the idea of limits. Also with many concepts in calc, my teacher did all the proofs, which was explained so that we can work with the proofs to get a general idea. Now, I feel like its turned around I have to prove, but I don't know why I'm having such a difficult time with proofs, maybe my first time. A student in my class has no problems with this, I don't know if this is first time encountering problems like this, but he was very pompous to say "he thinks he's more analytical" or something like that. Is it bad if I learn through teacher's example's and have to ask many questions to learn proofs if I see it for the first time? Can you guys give me a good advice or insights into my situation.

 

[ice109]

But that's what everyone says after they've learned a course. The fact of the matter is that the majority of students in school firstly, do not like maths, and secondly, are not very good at maths. So attempting to teach linear algebra to them would just be a complete waste of time and effort.

but we still make them take lots of math in high school ? i don't see your point?

and it's not the same thing. cal 3 could not be taught in high school because it requires 2 semesters of coursework prior to it, same for differential equations. analysis etc can't either because it requires calculus. linear algebra requires only algebra. there's absolutely no reason why you couldn't teach this class immediately or even concurrently with algebra, heck even pre algebra.

And there's even less chance of someone learning if you simply give them answers, is there?

so examples in books are pointless? I'm sure he has lots of other things to prove

contrary to what I'm sure lots of you think, i think it's absurd to expect students to do proofs, even the simplest ones, having never seen any.

antineutron: get a how to write proofs book.

An Introduction to Mathematical Reasoning by peter j eccles and

how to read and do proofs by solow

 

[mathwonk]

i learned from principles of mathematics by allendoerfer and oakley.

 

[Antineutron]

I'm going to order it now mathwonk, thanks

 

[cristo]

but we still make them take lots of math in high school ? i don't see your point?

What are you going to cut out of the syllabus in order to include linear algebra? Kids in high school have enough trouble expanding (x+1)^2, or using trigonometry. It will be a waste of time trying to teach them linear algebra. [By the way, our definitions of high school may be different; mine ends at 16, the age up to which it is compulsory to stay in school.]

and it's not the same thing. cal 3 could not be taught in high school because it requires 2 semesters of coursework prior to it, same for differential equations. analysis etc can't either because it requires calculus. linear algebra requires only algebra. there's absolutely no reason why you couldn't teach this class immediately or even concurrently with algebra, heck even pre algebra.

Great idea: let's teach kids number theory before we let them count, too

so examples in books are pointless? I'm sure he has lots of other things to prove

contrary to what I'm sure lots of you think, i think it's absurd to expect students to do proofs, even the simplest ones, having never seen any.

But if you tell someone what to do, then they do not have the chance to think about the question at all. In this case, all that was required was a counterexample. You could easily have hinted at this rather than doing it for him.

 

[ice109]

What are you going to cut out of the syllabus in order to include linear algebra? Kids in high school have enough trouble expanding (x+1)^2, or using trigonometry. It will be a waste of time trying to teach them linear algebra. [By the way, our definitions of high school may be different; mine ends at 16, the age up to which it is compulsory to stay in school.]

your definition not withstanding because high school doesn't end for most people at 16 so high school can't be defined like that. for all of your logic you're arguing a strawman. what i would have to cut of the curriculum is irrelevant to whether i could teach it or not. personally i would cut out a class called pathfinder that i was forced to take in high school that taught me that college existed and what i should do in applying. whether you agree or not is again irrelevant.

i don't know what kind of high school you went to but mine had math classes through calc 2
so obviously at least some of the students were comfortable with expanding (x+1)^2 and trig. if i recall correctly all high school students have to take 3 years of math up and to an including algebra two which is essentially college algebra. my contention being that that is all that is required for this course i think my argument is at least consistent.

Great idea: let's teach kids number theory before we let them count, too

again not the same thing considering the age at which we learn to count is before the abstraction process is fully developed and incidentally we do teach children modular arithmetic which is at least relevant to number theory very early. by high age most children's ability to abstract is developed and hence why we teach algebra and geometry then.

But if you tell someone what to do, then they do not have the chance to think about the question at all. In this case, all that was required was a counterexample. You could easily have hinted at this rather than doing it for him.

did you even read what i said? seems like you just restated what you said before. examples have their purpose. proving 1 specific set is not a vector space is the most important proof one can do.

here i'll make up for my indulgence

antineutron:

Show that the solution set of y = 2x+1 fails to be a vector space.

 

[mathwonk]

antineutron, i hope allendoerfer and oakley is to your liking, but honestly everyone finds their sweet spot on their own. i advise going to the library and perusing the various books on intro to proofs and just picking the one you like. good luck.

 

[Antineutron]

I'm sorry that this discussion turned into an argument. I just want to say you guys both have valid arguements and you guys are both right. Thanks for helping...

 

[mathwonk]

i recommend trying a simpler example than a vector space as practice. here's one:

define an incidence geometry to be a collection of points and lines satisfying the following three properties:
1) for each pair of distinct points, there is exactly one line containing them.
2) each line contains at least two points.
3) each point lies on at least two lines.

then : is a triangle an incidence geometry? (with vertices as points, and edges as lines).

what about a square? what about a tetrahedron? (always with vertices as points and edges as lines.)

prove there must be at least three points in an incidence geometry.
prove that if P is the number of points, then the number of lines is at most binomial coefficient "P choose 2". is this always the exact number of lines in an incidence geometry? when is it? if a tetrahedron in 4 space has 5 vertices, how many edges does it have? there is an incidence geometry with 7 points in which every line has exactly three points on it. how many lines are there?