In the book An introduction to manifolds by Tu, Loring W, there's an exercise:
Let $e_{1}, \ldots, e_{n}$ be a basis for a vector space $V$ and let $\alpha^{1}, \ldots, \alpha^{n}$ be its dual basis in $V^{\vee}$. Suppose $\left[g_{i j}\right] \in \mathbb{R}^{n \times n}$ is an $n \times n$ matrix. Define a bilinear function $f: V \times V \rightarrow \mathbb{R}$ by $$ f(v, w)=\sum_{1 \leq i, j \leq n} g_{i j} v^{i} w^{j} $$ for $v=\sum v^{i} e_{i}$ and $w=\sum w^{j} e_{j}$ in $V$. Describe $f$ in terms of the tensor products of $\alpha^{i}$ and $\alpha^{j}$, $1 \leq i, j \leq n$
And the answer says $$ f=\sum g_{ij}\alpha^i\otimes \alpha^j $$
I am getting confused about these notations. What is the relationship between $\alpha ^i \otimes \alpha ^j$ and $v^i w^j$? Is $\alpha ^i \otimes \alpha ^j$ equal to $v^i w^j$? Hope someone could help me with clarifying these notations.
For two linear functions $\gamma_1,\gamma_2 : V \to \mathbb R$ we define the bilinear map $\gamma_1 \otimes \gamma_2 : V \times V \to \mathbb R$ such that $(\gamma_1 \otimes \gamma_2)(v,w) = \gamma_1(v)\gamma_2(w)$. So, for any $(v,w) \in V \times V$ we have $$f(v,w) = \sum_{1 \leq i, j \leq n} g_{ij} v^{i} w^{j} = \sum_{1 \leq i, j \leq n} g_{ij} (\alpha^i \otimes \alpha^j)(v,w) = \bigg( \sum_{1 \leq i, j \leq n} g_{ij} \alpha^i \otimes \alpha^j \bigg)(v,w)$$ and then $$f = \sum_{1 \leq i, j \leq n} g_{ij} \alpha^i \otimes \alpha^j.$$