Spatial Algebra

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: $AB\ne BA$.

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

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

$p^{\prime }=Ap$

• $A$ is a transformation matrix
• $p$ is the original point
• and $p^{\prime }$ 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 “pipeline” of operations. For example, let’s say you first want to rotate a point and then translate it. You can represent this as:

$p^{\prime }=T(R(p))$ In matrix form, it becomes: $p^{\prime }=T\cdot (R\cdot p)=(T\cdot R)\cdot p$

• $T$ 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.