Linear mapping of a matrix to vector

Question

What does the statement "$\bf A(X) = b$", where $\bf X \in R^{m\times n}$ is the decision variable, $\bf A$ is the linear map from $\bf R^{m \times n}$ $\rightarrow \bf R^{p}$ and $\bf b \in R^p$ means. Since $\bf X$ is a linear map it is a $\bf m \times n$ matrix. What does $\bf A$ look like ?. What is the kind of linear operation applied on $\bf X$ to get $\bf b$.

Answer 1

Adding onto velut luna's answer: the linear map $A$ can be represented as $mnp$ coefficients $a_{ijk}$ with $$A(X)_k = \sum_{i=1}^m \sum_{j=1}^n a_{ijk}X_{ij}.$$ As you say, given a choice of basis for $\mathbb{R}^{m\times n}$ and $\mathbb{R}^p$, you can write $A$ as a matrix (as you can all linear transformations); however, there is no nice notation for $A$ so that it acts on $X_{n \times m}$ by matrix multiplication. You can "unroll" $X$ into a vector $\bar X$ of size $mn$, $$X = \left[\begin{array}{cccc}X_1 & X_2 & \cdots & X_m\end{array}\right], \bar X = \left[\begin{array}{c}X_1\\X_2\\\vdots \\ X_m\end{array}\right],$$ and now you can write $A$ as a $p\times mn$ matrix. I've also used, in my calculations, the following notation for $A(X)$ (note: not at all standard!!)

$$\left[\begin{array}{c} \left[\begin{array}{ccc}a_{111} & \cdots & a_{1m1}\\\vdots & & \vdots\\ a_{n11} & \cdots & a_{nm1}\end{array}\right] : \\ \vdots \\ \left[\begin{array}{ccc}a_{11p} &\cdots& a_{1mp}\\\vdots & & \vdots\\ a_{n1p} & \cdots & a_{nmp}\end{array}\right] : \end{array}\right]X$$ suggestive of the fact that the first entry of $A(X)$ should be computed by the Frobenius product $$\left[\begin{array}{ccc}a_{111} & \cdots & a_{1m1}\\\vdots & & \vdots\\ a_{n11} & \cdots & a_{nm1}\end{array}\right] :X,$$ etc.

Answer 2

I suppose you meant

"Since $A$ is a linear map it is a $m \times n$ matrix."

which is not the case.

$A$ is a linear map from an $mn$ dimensional vector space to a $p$ dimensional vector space. So it should be an $mn \times p$ matrix.