Modules with multiple operators

[cjellison]

Consider the set of 2x2 matrices which form a ring under matrix multiplication and matrix addition.

$\mathbb{R}^3$ is module defined over this ring.

So, we have three dimensional vectors whose elements are 2x2 matrices.

My question: Can I also define another "scalar multiplication" that is over the field of real numbers (well, I know you can)...what is such a structure called? For example, I want it to do the following:

$$3 \begin{pmatrix} \begin{pmatrix} a & b\\ c & d \end{pmatrix} & \begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}\\ \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & \begin{pmatrix} 1 & 2\\ 4 & 3 \end{pmatrix} \end{pmatrix} = \begin{pmatrix} 3 \begin{pmatrix} a & b\\ c & d \end{pmatrix} & 3\begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}\\ 3\begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & 3\begin{pmatrix} 1 & 2\\ 4 & 3 \end{pmatrix} \end{pmatrix} = \begin{pmatrix} \begin{pmatrix} 3a & 3b\\ 3c & 3d \end{pmatrix} & \begin{pmatrix} 3 & 3\\ 3 & 3 \end{pmatrix}\\ \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & \begin{pmatrix} 3 & 6\\ 12 & 9 \end{pmatrix} \end{pmatrix}$$

in addition to:

$$\begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \begin{pmatrix} \begin{pmatrix} a & b\\ c & d \end{pmatrix} & \begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}\\ \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & \begin{pmatrix} 1 & 2\\ 4 & 3 \end{pmatrix} \end{pmatrix} = \begin{pmatrix} \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \begin{pmatrix} a & b\\ c & d \end{pmatrix} & \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}\\ \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 2\\ 4 & 3 \end{pmatrix} \end{pmatrix} =\begin{pmatrix} \begin{pmatrix} a & b\\ c & d \end{pmatrix} & \begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}\\ \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & \begin{pmatrix} 1 & 2\\ 4 & 3 \end{pmatrix} \end{pmatrix}$$

 

[fresh_42]

Consider the set of 2x2 matrices which form a ring under matrix multiplication and matrix addition.

$\mathbb{R}^3$ is module defined over this ring.

Not automatically, so you have to define the module operation. How does a $2\times 2$ matrix operate on a three dimensional vector?

So, we have three dimensional vectors whose elements are 2x2 matrices.

This is another scenario, namely the vector space $\left( \mathbb{M}(2,\mathbb{R}) \right)^3$.

My question: Can I also define another "scalar multiplication" that is over the field of real numbers (well, I know you can)...what is such a structure called? For example, I want it to do the following:

$$3 \begin{pmatrix} \begin{pmatrix} a & b\\ c & d \end{pmatrix} & \begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}\\ \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & \begin{pmatrix} 1 & 2\\ 4 & 3 \end{pmatrix} \end{pmatrix} = \begin{pmatrix} 3 \begin{pmatrix} a & b\\ c & d \end{pmatrix} & 3\begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}\\ 3\begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & 3\begin{pmatrix} 1 & 2\\ 4 & 3 \end{pmatrix} \end{pmatrix} = \begin{pmatrix} \begin{pmatrix} 3a & 3b\\ 3c & 3d \end{pmatrix} & \begin{pmatrix} 3 & 3\\ 3 & 3 \end{pmatrix}\\ \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & \begin{pmatrix} 3 & 6\\ 12 & 9 \end{pmatrix} \end{pmatrix}$$

in addition to:

$$\begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \begin{pmatrix} \begin{pmatrix} a & b\\ c & d \end{pmatrix} & \begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}\\ \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & \begin{pmatrix} 1 & 2\\ 4 & 3 \end{pmatrix} \end{pmatrix} = \begin{pmatrix} \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \begin{pmatrix} a & b\\ c & d \end{pmatrix} & \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}\\ \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 2\\ 4 & 3 \end{pmatrix} \end{pmatrix} =\begin{pmatrix} \begin{pmatrix} a & b\\ c & d \end{pmatrix} & \begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}\\ \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & \begin{pmatrix} 1 & 2\\ 4 & 3 \end{pmatrix} \end{pmatrix}$$

No problem, it is a vector space, i.e. an $\mathbb{R}-$module.