# Change of Basis

Be VERY CAREFUL

Matrix multiplication is associative, but not commutative.

https://chat.openai.com/share/544dedb7-d4dd-4719-b304-c7e2fbeb3400

In the context of transformations, the order in which operations are applied is crucial. Suppose we have two matrices $A$ and $B$ that represent two different transformations. Multiplying these matrices in different orders will generally yield different results: $ABî€=BA$.

Although matrix multiplication is associative $A(BC)=(AB)C$, this does not mean that it is commutative $ABî€=BA$ in general.

In many applications like computer graphics or robotics, transformations are often represented in â€ścolumn-majorâ€ť form. In such a setup, points are column vectors and are transformed by right-multiplying them with transformation matrices:

$p_{â€˛}=Ap$

- $A$ is a transformation matrix
- $p$ is the original point
- and $p_{â€˛}$ is the transformed point

When you multiply on the RHS, you can intuitively read the series of transformations from right to left, giving you a â€śpipelineâ€ť of operations. For example, letâ€™s say you first want to rotate a point and then translate it. You can represent this as:

$p_{â€˛}=T(R(p))$ In matrix form, it becomes: $p_{â€˛}=Tâ‹…(Râ‹…p)=(Tâ‹…R)â‹…p$

- $T$ is the translation matrix and
- $R$ is the rotation matrix

Notice that we right-multiplied $p$ by $R$ and then right-multiplied the result by $T$, preserving the desired sequence of transformations.