What is a Vector: Sadri Hassani's Maths Physics Reading

[tobor8man]

Reading in Mathematical Physics by Sadri Hassani. It defines a vector abstractly. I will repeat that definition here rather more informally.

There are these things called vectors, a, b, x etc., that have these properties:

You can add them
a + b = b + a
a + (b + c) = (b + a) + c
a + 0 = a, 0 is the zero vector
a + (- a) = 0

You can multiply them by complex numbers (scalars) like c, d
c(d a) = (cd)a
1 a = a

Multiplication involving vectors and scalars is distributive
c(a + b) = c a+ c b
(c + d) a = c a+ d a

And that is it.

Given that definition, a scalar is a vector, a matrix is a vector, a tensor is a vector. Yes?

Mind you, I have also read that scalars and vectors are a kinds of tensors, of rank 0 and 1 respectively. True?

Am I confused? Should I be?

 

[Marco_84]

Reading in Mathematical Physics by Sadri Hassani. It defines a vector abstractly. I will repeat that definition here rather more informally.

There are these things called vectors, a, b, x etc., that have these properties:

You can add them
a + b = b + a
a + (b + c) = (b + a) + c
a + 0 = a, 0 is the zero vector
a + (- a) = 0

You can multiply them by complex numbers (scalars) like c, d
c(d a) = (cd)a
1 a = a

Multiplication involving vectors and scalars is distributive
c(a + b) = c a+ c b
(c + d) a = c a+ d a

And that is it.

Given that definition, a scalar is a vector, a matrix is a vector, a tensor is a vector. Yes?

Mind you, I have also read that scalars and vectors are a kinds of tensors, of rank 0 and 1 respectively. True?

Am I confused? Should I be?

you shouldn't be confused... you'll learn that first) ther's not only euclidean geometry----> different metrics----> then a vector is a generalization of a tensor, you'l learn the meaning of covarianca and controvariance... and you'll understand that everything id defined by its tranformation rule...

say an euclidean vector V is something that |V|^2 is an invariant under SO(3) and that under rotation transform as: V'^{\mu}=R_{\mu\nu}V^{\nu} where R belongs to SO(3).
keep going...
regards
marco

 

[slider142]

A vector is an element of a vector space. Thus, elements of the real line are vectors if you endow the real line with the algebraic structure of a vector space. The set of all continuous functions over R is also a vector space, where the vectors are functions. Tensors with the usual addition are also elements of a vector space when regarded in that manner. This is a space where the vectors are tensors, and so forth.