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  • https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/09%3A_Change_of_Basis/9.01%3A_The_Matrix_of_a_Linear_Transformation
    The fact that T=C1DTACB 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=C1DTACB 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 C1D).
  • https://math.libretexts.org/Bookshelves/Linear_Algebra/Book%3A_Linear_Algebra_(Schilling_Nachtergaele_and_Lankham)/10%3A_Change_of_bases/10.01%3A_Coordinate_Vectors
    which maps the vector vV 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 vV 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.

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