Coordinate Vector - Basis Transformation Matrix

Basis Transformation Matrix

Let B and C be two different bases of a vector space V, and let us mark with the matrix which has columns consisting of the C representation of basis vectors b1, b2, ..., bn:

_{C}^{B} = \begin{bmatrix} \ _C & \cdots & _C \ \end{bmatrix}

This matrix is referred to as the basis transformation matrix from B to C, and can be used for transforming any vector v from a B representation to a C representation, according to the following theorem:

If E is the standard basis, the transformation from B to E can be represented with the following simplified notation:

where

and

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