How to write tensor product in terms of normal addition and mulitplication? Im struggling to understand tensor product since there seems to be no way of wrting them as some combination of simple addition and multiplication. In the case of dot and cross product, such representation makes things a lot easier. so can we write tensor product in such a way. If not, why?
 A: In view of your comment on the answer from AOrtiz, I'm guessing that the question was intended to be about expressing the components of a simple tensor $v\otimes w$ in terms of the components of $v$ and of $w$. For that purpose, it is necessary to observe first that, if $V$ and $W$ have dimensions $m$ and $n$, then $V\otimes W$ has dimension $mn$. In more detail if the components of $v$ are $v_i$ ($1\leq i\leq m$) and the components of $w$ are $w_\alpha$ ($1\leq\alpha\leq n$), then the components of $v\otimes w$ are indexed by pairs $(i,\alpha)$. In those terms, there's a very simple formula: The $(i,\alpha)$-component of $v\otimes w$ is just the product $v_iw_\alpha$.
A: Edit: It is very challenging to give a precise description of what a tensor product is if you have never heard of vector spaces or know what a bilinear function is. Loosely speaking, the tensor product $T$ of two vector spaces is itself another vector space. $T$ is spanned by elements of the form $v\otimes w$. It is impossible to be more precise about what the elements $v\otimes w$ actually "are" without appealing to quotient spaces and the actual construction of $T$ from two simpler vector spaces.
Assuming you know what a bilinear function is, you can give another description of $v\otimes w$: given a bilinear map $B:V\times W\to L$, $v\otimes w$ is the element that is mapped by the function $\Phi:T\to L$ guaranteed in the universal property of the tensor product  to $B(v,w)$.
Again, these words won't make much sense unless you are already comfortable with vector spaces and bilinear functions.

The tensor product of two vector spaces $V$ and $W$ over a field $F$ is a vector space $V\otimes_F W$ whose elements are sums of so-called "simple tensors" of the form $v\otimes w$. On the other hand, the cross and dot products are examples of functions defined on Euclidean spaces. 
The distinction is that the tensor product is not an operation on a vector space whereas the dot and cross products are.
A: A much simplified working definition that may assist you:
Consider a finite $n$-tensor as a grid of values indexed by $n$ natural numbers. For example, a $2\times3\times2$ 3-tensor $t$ might have $1.5$ at $(2,1,1)$, in which case we'd write $$t(2,1,1) = 1.5$$
In this simple paradigm, consider an example tensor product between $p_{ace}$ (a $2\times 3\times 5$ 3-tensor), $t_{ab}$ (a $2\times 8$ 2-tensor), and $q_{cef}$ (a $3\times 5\times 2$ 3-tensor), that is $$p_{ace} t_{ab} q_{cef}$$
Find all the repeated indices, which happen to be $a,c,e$.
Find all the non-repeated indices, which are $b,f$.
Then the product is a tensor indexed by the non-repeated indices, and each of its elements is the sum of of all products over the repeated indices.
Hence if $$s_{bf} = p_{ace} t_{ab} q_{cef}$$ then $$s(b,f) = \sum _{a = 1}^{2}\sum _{c = 1}^{3} \sum _{e = 1}^{5} p(a,c,e)\cdot t(a,b)\cdot q(c,e,f)\;\;\forall b = 1\ldots 8, \; f = 1\ldots 2$$
If you consider an $r\times c$ matrix to be a 2-tensor $M_{rc}$, then the matrix product $C = AB$ you know and love is simply $$C_{ac} = A_{ab}B_{bc}$$ or equivalently $$C(a,c) = \sum _{b} A(a,b)\cdot B(b,c)\;\;\forall\; a,c$$
Hopefully this helps.
