How can I tell if this is a vector space?

[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 arguments 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?

 

[mathwonk]

hello?? is this helping?

 

[ice109]

you know mathwonk for someone who is apparently so knowledgeable you're pretty clueless. i weep for your students