What are the various branches of Vectors?

Question

I just wanted to begin learning 'Vectors'. But I am completely confused where I would have to start! There is vector Algebra, Vector geometry, Vector Analysis and what not.

So, I want to know what are all the most widely used branches of vectors in existence.

PS:- I am a 'wanna-be' Physicist. I just realized vectors are so important for Physics and so I want to know it. So I would also be obliged if you could tell where a noob would have to begin with his study of vectors. I would be heavily indebted to you if you could also suggest a free downloadable ebook on the subject. (I suppose my last 2 requests are not so on topic and so you can answer to those only if you wish to!)

Answer 1

In the first place, a vector is an element of a vector space.

A vector space is an environment (a set) where it makes sense

• to add any two elements $x$ and $y$, producing $x+y$,

and

• to scale any element $x$ with arbitrary real (or complex) scalars $\lambda$, producing $\lambda x$,

such that reasonable rules, like $$1\> x=x,\quad x+y=y+x,\quad \lambda(x+y)=\lambda\>x+\lambda\>y,\quad{\rm etc.},$$ hold.

Given that vector space is such a general notion there are all sorts of environments that have the structure of a vector space, and all sorts of things that can be called "vectors": triples, polynomials, continuous functions $f:\ [0,1]\to{\mathbb R}$, translations of euclidean $d$-space, equivalence classes of directed segments $[a,b]$, what have you.

Many of the vector spaces appearing in mathematics have additional structure elements and corresponding qualifying names: a norm ("normed space"), a scalar product ("euclidean space"), a "cross product" (euclidean $3$-space), a "pointwise product" (function spaces), a "convolution", and so on.

The algebraic, geometric, and computational theorems valid in "finitely generated" (= finite-dimensional) vector spaces are dealt with in linear algebra, the laws valid in connection with "vector-valued functions" and differentiation are dealt with in vector analysis, and so on.

What I'm saying is: There is no single book telling it all about vectors. "Vectors" are a façon de parler. The essential thing is that you intuitively accept a thing being a "vector" when told to do so.

Answer 2

A good start would be to pick up a good linear algebra book. Euclidean vectors, which is what you'll first see in physics as being called "vectors", are only one way of treating vectors. The concept is much more general and abstract, and much more powerful (as I joked with this parody of Magritte's "The Treachery of Images")

The choice of literature depends on your level. A good introduction is "Introduction to Linear Algebra" by Gilbert Strang, who also has his MIT lectures available at MIT Open Courseware. This is a a more practical and intuitive take on the subject which I think is a good thing.

Serge Lang's "Linear Algebra" is a very common linear algebra textbook in undergraduate courses, and it's a good comprehensive book for the core theorems with a more formal and abstract take. To supplement the formalism, I suggest taking up "Linear Algebra Done Right" by Sheldon Axler.

With that formalism in your belt, then you can start seeing vectors as a more general tool, something that will be very important later on if you decide to study physics formally and study analytical and quantum mechanics.

Good luck!