The fact that T=C−1DTACB means that the action of T on a vector v in V can be performed by first taking coordinates (that is, applying CB to v), th...The fact that T=C−1DTACB means that the action of T on a vector v in V can be performed by first taking coordinates (that is, applying CB to v), then multiplying by A (applying TA), and finally converting the resulting m-tuple back to a vector in W (applying C−1D).
which maps the vector v∈V to the n×1 column vector of its coordinates with respect to the basis e. The column vector [v]e is called the coordinate vector of v with respec...which maps the vector v∈V to the n×1 column vector of its coordinates with respect to the basis e. The column vector [v]e is called the coordinate vector of v with respect to the basis e. Note also that the map [⋅]e is an isomorphism (meaning that it is an injective and surjective linear map) and that it is also inner product preserving.