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

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

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

 Quote by AdrianZ now a dual space is the space spanned by that basis?
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 $f(v)=f(v^i e_i)=v^if(e_i)=\phi^i(v)f(e_i)$. Since v was arbitrary, this means that $f=f(e_i)\phi^i$. Since f was arbitrary, this means that $\{\phi^i\}$ spans V*.
 P: 320 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?

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