Inner product and the choice of coordinate frames

Question

How do I prove that dot product of two free vectors ( inner product in 2D and 3D vectors) does not depend on the choice of frames in which their coordinates are defined?

Answer 1

Since I guess it's homework, I'll just give a hint. A change of basis (or 'frame') is a linear mapping, i.e. applying an certain matrix $A$, sending each vector $v \mapsto A.v.$ What properties does $A$ have?

Answer 2

The dot-product is in the first place a bilinear function of two vector variables with the additional property that ${\bf x}\bullet{\bf x}>0$ for ${\bf x}\ne{\bf 0}$. For any basis $({\bf e}_i)_{1\leq i\leq n}$ of your space the dot-product is encoded in a certain symmetric matrix $G:=(g_{ik})_{1\leq i\leq n,1\leq k\leq n}$. This matrix changes with the basis in a characteristic way. When a dot product is given then there are certain distinguished bases, namely the orthonormal ones. For these the matrix $G$ is just the identity matrix, and in the corresponding coordinates the dot-product computes as $(*)\ {\bf x}\bullet{\bf y} =\sum_{i=1}^n x_i y_i$. In particular, if one starts with the "standard space" ${\mathbb R}^n$ then the standard basis is considered orthonormal, whence one has the standard formula $(*)$ there.