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:

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

in addition to:

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

 

[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:

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

in addition to:

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

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