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Change of Basis

A matrix can be seen as a linear transformation of a space. \mathbf{Ax} means transform the vector \mathbf{x}, which is in the original space, into the new space, but still using the original basis for coordination. The original basis means the perpendicular unit vectors. In this case, \mathbf{x} should be a vector represented in the original basis coordination system.

Let's consider another problem, \mathbf{x} is (3, 2). However, it is not in the original coordinate system but a new coordinate system, which uses \mathbf{A}'s column vectors as basis. 3 and 2 are the scalars for the column vectors in \mathbf{A}. How can we translate the vector \mathbf{x} back to the original coordination system? I mean what is the coordinate of the \mathbf{x} represented by the original coordination system. It is the same as the above example, it should be \mathbf{Ax}.

So a matrix can have different meaning's in different situations. It can either mean a transformation of the space, or a translation from one coordination system to another. It depends on the meaning of the vector multiplied on the right. If x is using the new space basis, then means translation. If in the old one, it means transformation.

Another problem is how to translate a vector \mathbf{x} represented in the original system into the new coordination system. Just use \mathbf{A}^{-1}\mathbf{x}, where \mathbf{A} consists of the basis vectors of the new coordination system as columns. The reason is as follows. Suppose \mathbf{x}$ is using the new coordinate system specified by \mathbf{A}. \mathbf{A}^{-1}\mathbf{Ax} is still \mathbf{x}. \mathbf{v} = \mathbf{Ax} is the translated \mathbf{x} in the original coordinate system. So \mathbf{v} is \mathbf{x} described by the original coordination system. \mathbf{A}^{-1}\mathbf{Ax}=\mathbf{x} \Rightarrow \mathbf{A}^{-1}\mathbf{v}=\mathbf{x}, so \mathbf{A}^{-1} can translate \mathbf{v} to \mathbf{x}. So \mathbf{A}^{-1} is the opposite translation of \mathbf{A}.

If we want to do an operation (rotate a vector for 90 degree) on a vector in the new coordination system, what we do is translate it to the old system, do the operation, translate it back to the new. \mathbf{A}^{-1}\mathbf{BAx}, \mathbf{B} is the operation matrix, \mathbf{A} is the translation matrix.