This problem was left as an exercise in algebra class of mine but I am not able to make any progress. I have been following book by atiyah and macdonald and I must say I had difficulty in understanding the concept of tensor products but I managed to by reading the part on tensor products in Atiyah and Macdonald many times and also watching video here:https://www.youtube.com/watch?v=kGkOo7w8xeM
But I am still not able to proceed on this problem.
Problem: Let $e_1 , e_2 ,\dots, e_n$ be a basis for a vector space V and $\alpha^{1},\dots,\alpha^{n}$ be its dual basis in $V^{*}$. Suppose that $(g_{ij})$ is an $n \times n$ matrix. Define a bilinear function $ f\colon V\times V \to \mathbb{R}$ by $$f(v,w) = \sum_{1\leq i, j\leq n} g_{ij} v^i w^j$$ for $v= \sum_i v^i e_i $ and $w= \sum_j w^j e_j$. Describe $f$ in terms of the tensor products of $\alpha^i$ and $\alpha^j$ , $1\leq i,j\leq n$.
Can you please show me how should I attempt this problem?
Simply observe the following. The map $f$ is defined by $$f(v,w) = \sum_{i,j} g_{ij}v^iw^j = \sum_{i,j} g_{ij}\alpha^i(v)\alpha^j(w) = \sum_{i,j}g_{ij}(\alpha_i \otimes \alpha_j)(v,w).$$ The identities hold as $$(\alpha_i \otimes \alpha_j)(v,w) := \alpha_i(v)\alpha(v_j) = \sum_{k,l} v^kw^l \alpha_i(e_k)\alpha_j(e_l) = \sum_{k,l} v^kw^l \delta_{ik}\delta_{jl} = v^iw^j,$$ where the $\delta_{ij}$s are Kronecker deltas. Hence $f = \sum_{i,j} g_{ij} \alpha_i \otimes \alpha_j$.