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I know I can write scalar product between $v$ and $w$ vectors in matricial form as $v\cdot w=v^TMw$.

I have a not canonical base.. my doubt is: in the formula, coords of $v$ and $w$ are meant to be related to canonical base or to my base?


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We don't particularly have canonical bases for general vector spaces. In this sense, every basis is as good as any other basis! (Yes, you would like to distinguish between eigenbases and orthogonal bases, but those are special). In this case, if you're given a basis, everything should be in terms of that basis, because your vector space can't normally distinguish between other bases in the first place. – Calvin Woo Sep 20 '12 at 19:20
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I understand your question in the way that your vetor space has some underlying inner product $(\cdot, \cdot)$, and that you have a basis $\{\mathbf{e}_k\}$, which is known but not orthonormal in the inner product, $$ \mathbf{v} = \sum_k v_k e_k, \quad \mathbf{w} = \sum_k w_k e_k. $$ Then $$ (\mathbf{v}, \mathbf{w}) = \sum_k \sum_\ell v_k (e_k,e_\ell) w_\ell = v^\top M w , $$ where $$ M_{k\ell} = (\mathbf{e}_k,\mathbf{e}_\ell). $$

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