I’m a self-taught math student and i want to learn how to work and calculate with tensors. I’m pretty familiar with the theory behind them, but something about the the different notations just does not sink in. I thought of an experiment that might help me understand it better, and i would really appreciate your help. The experiment is very straightforward:
Suppose i have a simple finite field extension $$K=\frac{\mathbb{Q}(\alpha)}{\mathbb{Q}}$$ If $\vec{a}$ and $\vec{b}$ are in $K$, then, for some natural number $d$, one can write: $$\vec{a}=a_0+a_1\alpha+\cdots+a_d\alpha^d$$ $$\vec{b}=b_0+b_1\alpha+\cdots+b_d\alpha^d$$ with $\{a_0;\cdots;a_d;b_0;\cdots;b_d\}\in\mathbb{Q}$. A typical multiplication in $K$ would go like this: $$\vec{a}\cdot\vec{b}=\sum_{i+j\leq 2d}a_ib_j\alpha^{i+j}$$ But this is only the sum of all products, and all products are contained in the components of a tensor product of two vectors; so it’s clearly some contraction (or multilinear form… please, adress the difference in the answer) on the 2-tensor $\mathcal{T}\in K^{\otimes 2}$: $$\mathcal{T}=\vec{a}\otimes\vec{b}$$ All this said, my question is the following: how can i express $\vec{a}\cdot\vec{b}$ in the language of tensors? If that is too broad, how can i express that product like a tensor contraction or multilinear form?
Well, after some time looking in forums and papers, i found something that could be a proper answer:
Let $V$ be the matrix vector space: $$V=K\otimes K$$ And $*:K^2\rightarrow V$ the outer product of vectors, we then write: $$\vec{a}*\vec{b}=\begin{pmatrix} a_0b_0 & a_0b_1 & \cdots & a_0b_d \\a_1b_0 & a_1b_1 & \cdots & a_1b_d \\ \vdots& \vdots &\ddots & \vdots \\a_db_0 & a_db_1 & \cdots & a_db_d \end{pmatrix}$$ The basis of $V$ as a tensor product is $$B=\begin{pmatrix} 1 & \cdots &\alpha_d \\\vdots &\ddots & \vdots \\\alpha_d &\cdots & \alpha_{2d} \end{pmatrix}$$ We can then define the product in k as the inner product in $V$: $$\vec{a}\cdot\vec{b}=\langle\vec{a}*\vec{b},B\rangle_F$$