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What is a vector?

  1. Mar 9, 2008 #1
    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?
  2. jcsd
  3. Mar 9, 2008 #2
    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: [tex]V'^{\mu}=R_{\mu\nu}V^{\nu}[/tex] where R belongs to SO(3).
    keep going....
  4. Mar 9, 2008 #3
    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.
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