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Teaching Vectors Properly (For Everyone) - Page 9

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decerto
Profile Joined November 2011
244 Posts
Last Edited: 2012-02-05 11:05:10
February 05 2012 10:39 GMT
#161
On February 05 2012 19:11 Jombozeus wrote:
I am currently taking Calc B in my college freshman year, and I've taken regular math in highschool with only introduction to vectors. So I am what you would call a stereotypical -meh- math student that would be learning vectors.

I read through your explanation of vectors 3 times and I have not a single fucking clue what its saying.

Which I guess proves you wrong.


Physics vectors and real numbers 17, (1,2) or (3,4,5) are just a special case of vectors called n-tuples with n being the dimension. They are the vectors of vector spaces R, R^2, R^3 which are called euclidean vector spaces after the famous mathematician Euclid who developed the geometry in these spaces in his book "Elements".

Vector spaces are just sets of elements like the real numbers with additional structure, like how to combine them etc(that big list of things he wrote).

These elements can also be functions, complex numbers and polynomials etc so long as they obey those properties he wrote. And because they do obey those property's it gives rise to a lot of new property's that can be derived like the inner product etc.

And just like sets you can have subsets(smaller sets that are contained in the original set) which when they obey these property's they are subspaces

Linear algebra is just the study of these Vector spaces and their subspaces and mappings between vector spaces which preserve linearity(maps a line to a line).

But Im curious if vectors are just a further special case of tensors?, only In first year so I havent come across them properly
rabidch
Profile Joined January 2010
United States20289 Posts
February 05 2012 10:42 GMT
#162
when i see average high school students being taught the way as outlined as OP and the majority of them understanding it quickly, i'll agree with the OP. otherwise not.
LiquidDota StaffOnly a true king can play the King.
Lpspace
Profile Joined April 2011
4 Posts
February 05 2012 14:12 GMT
#163
On February 05 2012 17:46 paralleluniverse wrote:

Show nested quote +
This is one example why I think you shouldnt try to teach. If you want to explain fundamentals you have to get it right otherwise you are hurting the understanding more than you help it in the long run.


No shit. That's the whole point of this post.

The very first picture depicts a complex number as a vector.

Yet, students get it drilled into them that you cannot multiply 2 vectors (which is correct), but then why can you multiply two complex numbers if they are vectors? The answer requires an understanding of vector spaces, and that when complex numbers are presented as a vector over the reals, there is no vector multiplication, so complex multiplication isn't vector multiplication. But when complex numbers are presented as a field, the field multiplication is complex multiplication.


Well if that's the point, here's an example issue from your posts. The reason you can multiply two complex "vectors" together is not that the complex numbers form a field. It is because the complex numbers form an algebra over the reals. There are plenty of other interesting vector spaces which you can extend to algebras but do not have a field structure. For example, look at the multiplication defined for lie algebras which gives rise to a whole lot of "zero divisors".
zag
Profile Joined September 2010
Australia60 Posts
Last Edited: 2012-02-05 14:38:14
February 05 2012 14:29 GMT
#164
edit: was plainly wrong sry
Envy is the highest form of recognition
Caloooomi
Profile Joined July 2010
Scotland188 Posts
February 05 2012 14:35 GMT
#165
I like it. Started off learning vectors and didn't grasp it as well as other maths topics which ended up hurting me in the long run!
Booga booga booga~
gyth
Profile Blog Joined September 2009
657 Posts
February 05 2012 15:09 GMT
#166
You should only need to explain enough to understand the content and not teach something that is wrong

Physics teaches the difference between scalars and vectors. That can't be done by using a mathematically rigorous definition of a vector.


Complex numbers are often introduced as vectors.

Introduced as the same special case R^2 vector used in physics. It is almost like teachers got together and agreed upon how vectors should be taught outside of university level linear algebra courses.
The plural of anecdote is not data.
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