What topics from linear algebra do I need to study tensors?

AdrianZ

Yaayy! I guess I've understood what a dual basis is. now a dual space is the space spanned by that basis? If yes then I've understood what a dual space is.

now, what are co-variant and contra-variant vectors? How do they arise in math/physics?

Fredrik

In the simplest non-trivial case [itex]V=\mathbb R^2[/itex], [itex]\phi^i(e_j)=\delta^i_j[/itex] are the four equalities [tex]\begin{align}
\phi^1(e_1) &=1\\
\phi^1(e_2) &=0\\
\phi^2(e_1) &=0\\
\phi^2(e_2) &=1
\end{align}[/tex] Since the [itex]\phi^i[/itex] are assumed to be linear, we have [itex]\phi^i(v)=\phi^i(v^je_j)=v^j\phi^i(e_j)=v^j\delta^i_j=v^i[/itex], and in particular [itex]\phi^1(ae_1+be_2)=a\phi^1(e_1)+b\phi^1(e_2)=a[/itex].

I explained that in the post I linked to earlier. The one that made you say that you didn't understand the OP's question.

Yes, but that would be a weird way to define it. V* is the set of linear functionals from V into ℝ. Addition and scalar multiplication are defined in the obvious ways:

(f+g)(v)=f(v)+g(v)
(af)(v)=a(f(v))

These definitions give V* the structure of a vector space. Let f in V* and v in V be arbitrary. We have [itex]f(v)=f(v^i e_i)=v^if(e_i)=\phi^i(v)f(e_i)[/itex]. Since v was arbitrary, this means that [itex]f=f(e_i)\phi^i[/itex]. Since f was arbitrary, this means that [itex]\{\phi^i\}[/itex] spans V*.

AdrianZ

great. now I have almost understood the idea behind defining dual spaces and also the idea behind covariant and contravariant vectors. your post about co/contra-variant vectors was a great one.

now the only thing remained unclear for me is that I want to see some mathematical/physical examples of tensors. and I want to understand how we can interpret different physical situations with the new language I've learned. I have almost 0 level of knowledge about differential geometry, so don't go into very advanced topics from physics please (like GR).

Is there any other thing that I need to know about tensors?