I am told that for two vectors $v,w$ in a vector space $V$ $v \cdot w = \underline{v}^T \underline{w}$ only in an orthonormal basis. This is clear if the basis is the standard $(1,0,\dots,0), (0,1,\dots,0),\dots $ etc but surely when this basis changes (and is still orthonormal) the two vectors $v,w$ will still be the same and so $v \cdot w$ won't change but the coordinates $\underline{v}, \underline{w}$ with respect to that basis will change, so $\underline{v}^T \underline{w}$ will also change?
Also, for a Euclidean space, where you have a vector space and a positive definite symmetric bilinear form, what is the 'dot product'? Is it even definable if you don't know what the basis is?
I know these two questions might be slightly contrasting but I can't get my head around the relation between dot product and bilinear forms!
Thanks!