Given a linear map T:V → W between vector spaces, and a choice of bases 𝒱 for V and 𝒲 for W, then the matrix of the linear map T sends the coordinate vector of v ∈ V to the coordinate vector of Tv in W. Denote this matrix as _𝒲T_𝒱. If 𝒱’ and 𝒲’ are two other bases, then _𝒲’T_𝒱’ = (_𝒲’ I _𝒲)(_𝒲 T _𝒱)(_𝒱 I _𝒱’), where I denotes the identity map (on V or W). The matrices _𝒲’ I _𝒲 and _𝒱 I _𝒱’ are referred to as change of basis matrices. Respectively, they change the coordinates of a vector with respect to 𝒲 (or 𝒱’) to the coordinates of the same vector with respect to 𝒲’ (or 𝒱).