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Linearization a Bilinear form
I’m currently trying to linearize a bilinear form that’s defined on $V\times V$ where $V$ is a finite-dimensional vector space.
I’m reading over some notes and to do this, i’m given a basis $\{e_1, \ldots, e_n\}$ for $V$.
I’m trying to prove that the formula for linearizing a form is
$$\begin{array}{cccc}
\beta\left(X(u+v),w\right) = \sum_{j=1}^{n} \beta_{ij}(u)\beta_{jk}(v)w_{k} \hspace{1cm} \forall u,v,w\in V \\
\end{array}$$
where $\beta_{ij}(u)$ denotes $\beta(e_i,u)$ and $e_i,u$ and $v,w$ is the $i^{th}$ and $k^{th}$ basis vector respectively.
However, when trying to simplify the derivation for the linearizing formula for a bilinear form I find this to be wrong:
$$ \begin{array}{cccc}
\beta\left(X(u+v),w\right) = \sum_{i=1}^{n} \beta_{ii}(u)\beta_{ii}(v)w_{i} + \sum_{j,k=1}^{n} \beta_{ij}(u)\beta_{kj}(v)w_{j} \\
\end{array}$$
I was a bit confused as to why the derivation for the linearizing formula for a bilinear form
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