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Let $E = \{e_1, ..., e_n\}$ - base of the $V$, $A: E \times E \to K$ - any transformation. Then there is exactly one bilinear form $B: V \times V \to K$ such that $B (e_i, e_j) = A (e_i, e_j)$ for each $e_i, e_j \in E$.

How do you prove it?

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Can I write it so?:

Let $A(e_{i},e_{j})=a_{ij}$ - any transformation. Since $B(e_{i},e_{j}) = A(e_{i},e_{j})$, so $A(x,y)=\sum_{i, j = 1}^{n}a_{ij}x_{i}y_{j}=B(x,y)$

which proves the uniqueness bilinear form $B:V \times V \rightarrow K$.

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This is basically right, except that $A$ is not defined on $V \times V$. But writing $x = \sum x_i e_i$ and $y = \sum y_j e_j$, then you are correct that $B(x,y) = \sum x_i y_j B(e_i, e_j) = \sum a_{ij} x_i y_j$. I broke up the argument into two steps - the first uses bilinearity, while the second uses that $B(e_i, e_j) = A(e_i, e_j)$. – Michael Joyce Jan 30 '13 at 15:25
Thank you very much ;) Let $E = \{e_{1}, ..., e_{n}\}$ - base of $V$, $A: E \times E \rightarrow K$ - any transformation, such that $A(e_{i}, e_{j}) = B (e_{i}, e_{j})$ for a bilinear form $B: V \times V \rightarrow K$. Then for any $x = \sum {x}_{i} {e} _ {i}$ and $y = \sum y_{j} {e}_{j}$ $A(x,y) = A( \sum {x}_{i} {e}_{i} \sum {y}_{j} {e}_{j}) = \sum x_{i} y_{j} A(e_{i}, e_{j}) = \sum x_{i} y_{j} B(e_{i}, e_{j}) = B(x,y)$ it's ok? – sigma123 Jan 30 '13 at 16:21
In your last equation, $A(x,y)$ is not defined for all $x,y \in V$. So the first two equalities don't make sense. The rest is fine. – Michael Joyce Jan 30 '13 at 17:28
I understand now. Once again thank you very much ;) – sigma123 Jan 30 '13 at 18:33

Given two vectors in $V$, expand each as a (unique) linear combination of the basis vectors, then use the bilinearity of $B$.

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