Inner Product and the Length of Vectors

An inner product of the Euclidean bases e,- and cj gives the following ortho-normal condition  

Note that the coefficient v,- of a vector v = v,- e,- is obtained by an inner product between v and e,-  

Assume a linear transformation A that maps a vector u into another vector v (i.e., V = Au) where the coefficients are represented by a matrix (Ay ). Then, an inner product between this v and a vector w is given as 

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