Question on the differences in the definitions of what a tensor is Below are the common definitions of tensor.
a. "a tensor is a quantity which transforms according to a definite law under the change of the coordinate system".
b. "a tensor is a multilinear function which takes vectors and duals and produces a scalar"
Questions:


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*How are these two definitions related? That is, how can we start with one of definitions and arrive at the other?

*What is the significance of being "multilinear"? If we have a function of vectors & duals producing a scalar which is not multilinear, what breaks down? 


The way I have so far understood a tensor is as follows. Multiple vectors might act on each point in space and produce a result. Tensor is a way of describing the whole phenomena. But again, I could not relate this to the definitions. Why "invariance under coordinate change" or "mulitlinearity" are required here?
 A: Let's start by making definition a detailed: under general coordinate transformations, a tensor satisfies$$T^{a_1\cdots a_p}_{\qquad b_1\cdots b_q}=\prod_{i=1}^p\frac{\partial x^{a_i}}{\partial y^{A_i}}\cdot\prod_{j=1}^q\frac{\partial y^{B_j}}{\partial x^{b_j}}\cdot T^{A_1\cdots A_p}_{\qquad B_1\cdots B_q}$$(all "products" herein are also contracted over). You can verify the contraction of two tensors is a tensor; you'll just need to use$$\frac{\partial x^a}{\partial y^A}\frac{\partial y^A}{\partial x^b}=\delta^a_b,\,\frac{\partial y^A}{\partial x^a}\frac{\partial x^a}{\partial y^B}=\delta^A_B.$$Therefore, tensors are closed under contracting away all the indices to make a scalar, viz.$$S:=T^{a_1\cdots a_p}_{\qquad b_1\cdots b_q}\prod_iD^{(i)}_{a_i}\prod_jV^{b_j}_{(j)}$$(our duals $D^{(i)}_{a_i}$ and vectors $V_{(j)}^{b_j}$ have bracketed labels that shouldn't be confused with their contractible indices). Then $S$ is invariant under GCTs, showing definition a implies definition b - including, in particular, the multilinearity in the $D$s & $V$s. We can prove the converse using scalars' invariance and the duals' and vectors' behaviour under GCTs,$$D_{a_i}^{(i)}=\frac{\partial y^{A_i}}{\partial x^{a_i}}D_{A_i}^{(i)},\,V^{b_j}_{(j)}=\frac{\partial x^{b_j}}{\partial y^{B_j}}V^{B_j}_{(j)}.$$
