Linear Algebra 1: my first class

[Lucas Ayres]

I just had my first Linear Algebra at the Universidade Federal de Minas Gerais (Brazil). The teacher talked about vector spaces, and I found it very abstract. I am starting to think about the subject, and any comments are welcome! I just need a place to vent my thoughts and frustrations, so I am creating this topic.

 

[Lucas Ayres]

Who was the first mathematician to coin the term "vector spaces"?

 

[Lucas Ayres]

Hmm, Wikipedia says it was Giuseppe Peano in 1888.

"Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva, Turin"

 

[jfgobin]

Hello Lucas,

Hang on! In no time, you'll grasp the concepts and everything will be clear.

If you want a visual picture of a vector space, imagine that you have something with a number of possible independent directions (the dimension of the space). Add to that the fact you can measure the distance between two points of the space, and that you have a number of axioms (listed here) that are verified.

Vector spaces are used since the mid 17th century, but as you noted, Peano was the first to give them an abstract treatment, probably following his work on the axiomatization of mathematics.

You can build vector space on any number field: for instance over the real numbers (that's the common vector space you are used to), but also over the rationals or the complex.

If I may offer you an exercise: next time you are out for a walk, look at everything around you, and imagine that you have arrows going from you to the various objects and people, with an arrow attach to them to denote their movement. Look at how you can combine these arrows, calculate distances and things.

As I said: hang on. With some thoughts and time, it will become clearer.

J.

 

[Lucas Ayres]

Thank you very much! Heading for my second class in about an hour...

 

[GoodSpirit]

It's pretty easy! You'll see

 

[NegativeDept]

Whenever I get confused with vector and linear algebra concepts, I start by calculating examples with low-dimensional real vectors.

Vectors don't have to be arrows in 2D or 3D space; that's the whole point of abstract vector space theory. But you can often get some intuition by drawing arrows in $\mathbb{R}^2$ or $\mathbb{R}^3$. For example, the triangle inequality is an axiom for defining a norm on a vector space. For the vector spaces $\mathbb{R}^2$ and $\mathbb{R}^3$, it just says that the long side of a triangle can't be longer than the sum of the short sides.

 

[dkotschessaa]

How'd it go? Or is it still going?

I really, really enjoyed linear algebra this year. I found myself doing extra problems for fun.

The abstract vector space stuff is a pain at first, and it's only a small part of the course (in our case), but then when you stop doing them you kind of miss them.

I'm studying abstract algebra right now on my own before taking the class in the summer, and they tail into each other nicely.

 

[HallsofIvy]

The crucial thing is learning the precise words of the definitions. It isn't enough to have a general idea of what a "vector space" or "basis" is. Definitions in mathematics are "working" definitions- you use the precise words in proofs and in solving problems.

 

[dkotschessaa]

The crucial thing is learning the precise words of the definitions. It isn't enough to have a general idea of what a "vector space" or "basis" is. Definitions in mathematics are "working" definitions- you use the precise words in proofs and in solving problems.

Yes. Our professor actually had us memorize definitions for the test. On the surface, this might kind of seem like old school, rote learning . But it was actually very helpful. He insisted on precision with regards to the notation. Even "0" isn't good enough - which 0? The zero in the original vector space or the target vector space? This wasn't pedantic - those symbols mean something.

I'm taking this same approach now in teaching myself abstract algebra. Obviously, you can blindly/rote learn definitions, but if you spend some time contemplating them it's not really memorization in that sense.

Methinks I will call them "contemplation cards" from now on instead of "flash cards."

-Dave K

 

[Lucas Ayres]

How'd it go? Or is it still going?

I really, really enjoyed linear algebra this year. I found myself doing extra problems for fun.

The abstract vector space stuff is a pain at first, and it's only a small part of the course (in our case), but then when you stop doing them you kind of miss them.

I'm studying abstract algebra right now on my own before taking the class in the summer, and they tail into each other nicely.

Unfortunately, I failed . I guess it was my fault. But I'll try it again next year! Never give up!

 

[dkotschessaa]

Unfortunately, I failed . I guess it was my fault. But I'll try it again next year! Never give up!

aww man. Sorry to hear. It happens. What do you think happened? Was it definitions? My professor was very straightforward about how important definitions were. It worked for Linear Algebra and I'm taking the same advice to heart in abstract algebra and doing very well.

-Dave K