Editing Coordinate vector (section)

== Basis transformation matrix ==
Let ''B'' and ''C'' be two different bases of a vector space ''V'', and let us mark with <math>\lbrack M \rbrack_C^B</math> the [[matrix (mathematics)|matrix]] which has columns consisting of the ''C'' representation of basis vectors ''b<sub>1</sub>,  b<sub>2</sub>, …, b<sub>n</sub>'': 
:<math>\lbrack M\rbrack_C^B = \begin{bmatrix} \lbrack b_1\rbrack_C & \cdots & \lbrack b_n\rbrack_C \end{bmatrix} </math>

This matrix is referred to as the '''basis transformation matrix''' from ''B'' to ''C''. It can be regarded as an [[automorphism]] over <math>F^n</math>.  Any vector ''v'' represented in ''B'' can be transformed to a representation in ''C'' as follows:
:<math>\lbrack v\rbrack_C = \lbrack M\rbrack_C^B \lbrack v\rbrack_B. </math>

Under the transformation of basis, notice that the superscript on the transformation matrix, ''M'', and the subscript on the coordinate vector, ''v'', are the same, and seemingly cancel, leaving the remaining subscript. While this may serve as a memory aid, it is important to note that no such cancellation, or similar mathematical operation, is taking place.

===Corollary===
The matrix ''M'' is an [[invertible matrix]] and ''M''<sup>−1</sup> is the basis transformation matrix from ''C'' to ''B''. In other words,
:<math>\begin{align}
      \operatorname{Id} &= \lbrack M\rbrack_C^B \lbrack M\rbrack_B^C = \lbrack M\rbrack_C^C \\[3pt]
  &= \lbrack M\rbrack_B^C \lbrack M\rbrack_C^B = \lbrack M\rbrack_B^B
\end{align}</math>