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.