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?
Thanks!
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?
Thanks!
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). $