Do tensors form a vector space?

Question

Also, in what sort of a math class would you discuss tensor mathematics in detail?

Answer 1

http://en.wikipedia.org/wiki/Tensor
http://mathworld.wolfram.com/Tensor.html

Answer 2

A tensor, loosly speaking, is anything that transforms one thing into another. A tensor can be either: a scalar, which does not form a vector space, a n- dimensional vector, an array of vectors of any dimension, called a matrix, or more complicated matrix-like structures. Scalars, vectors, vector spaces and matricies are studied in linear algebra, while higher-order tensors are studied in tensor analysis. The most important property of a tensor is its rank; the rank of a tensor dictates how it operates on other vectors or tensors. The tensor is written with a subscript which tells you its rank. Here are some examples of specific rank tensors,
Scalar- rank 0 tensor.
Vector- rank 1 tensor. A vector V is written with a subscript which denotes how many dimensions the vector has. For example, if a vector has a subscript 3, that means it has 3 dimensons; V sub 1 would denote the componant on the x-axis, V sub 2 on the y-axis, and so on.
Matrix- rank 2 tensor. Since a matrix is an array of vectors, it has two subscripts. Imagine if you had two vectors. You would need an extra subscript, either a one or a two, to tell you which vector you were talking about. As an example, V sub 1, sub 1 would mean the first componant of the first vector, V sub 1, sub 2 would mean the first componant of the second vector, and so on.
Higher rank tensors are more complicated and do not have specific names, although they are just as easy to understand and work with as lower rank tensors. Tensors are really easy once you get them; hope this helps!

Answer 3

yes indeed, Linear Algebra

I have seen that someone have told you that scalar is not a vector
ok here is the definition of a vectorspace:

1)v+w=w+v (commutative law of addition)
2)(v+w)+x=v+(w+x) (associative law of addition)
3)v+0=v (additive identity law)
4)there exist for every vector v an vector -v so
v+(-v)=n (additive inverse law)
5)r(v+w)=rv+rw distributive law)
6) (r+s)v==rv+sv (distributive law)
7) r(sv)=rs(v) associative law of multiplication)
8) 1v=v (scalar identity law)

r,s are scalars, v,w,x are vectors

a scalar is just an 1dimensional vector, but if you have a limitet set of numbers let say {1,2,3}, you ofcause have to prove that this set sartisfied the above given rules

let say v=(kn in R l - infinity Prove for v is a vector space:

1) k1+k2=k2*k1 true for all k1,k2 in R
2) (k1+k2)+k3=k1+(k2+k3) tru for all k1,k21,k3 i R
3) k1+0=k1 true for all k1 in R
4) k1+(-k1)= 0 true for all k1 in R
5) r(*(k1+k2)= rk1+rk2 true for all k1,k2 in R
6) (r+s)=rk1+sk2 true for all k1,k2 in R
7) r(sk1)=(rs)k1 true for all k1,k2 in R
8) 1*k1=k1 true for all k1 in R


so v is a vector space.