Can a Matrix A Equalize Vectors u and v in Different Reference Frames?

[taybasta]

Hi everyone,

Given two different reference frames in a vector space; say left and right. v is a vector defined in the left frame and u is a vector defined in the right frame.
What is the nature of a matrix A that can satisfy the equality u= A.v?
Thank you

 

[HallsofIvy]

If the vector space has dimension n, then A is represented by the n by n, non-singular, matrix whose columns are the coefficients of vectors in u when written as a linear combination of the vectors in v.

 

[taybasta]

Thanks HallsofIvy,
the problem is that the two vectors u and v are not defined in the same reference frame.