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Change of Coordinates Matrix

A change of coordinates matrix, also called a transition matrix, specifies the transformation from one vector basis to another under a change of basis. For example, if B={u,v} and B^'={u^',v^'} are two vector bases in R^2, and let [r]_B be the coordinates of a vector r in R^2 in basis B and [r]_(B^') its coordinates in basis B^'.

Write the basis vectors u^' and v^' for B^' in coordinates relative to basis B as

[u^']_B=[a; b]
(1)
[v^']_B=[c; d].
(2)

This means that

u^'=au+bv
(3)
v^'=cu+dv,
(4)

giving the change of coordinates matrix

P=[a c; b d]
(5)

which specifies the change of coordinates of a vector r under the change of basis from B^' to B via

[r]_B=P[r]_(B^').
(6)

See also

Basis Vector, Change of Basis, Vector Basis

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Cite this as:

Weisstein, Eric W. "Change of Coordinates Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ChangeofCoordinatesMatrix.html

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