Is there a difference between a matrix as a transformation and a matrix as a set of column vectors In linear algebra a matrix is a representation of a transformation. But people also use matrices for storing column vectors (for example think about the data matrix as input to PCA). Can we think of the latter also as a transformation or these two notions of matrices are totally different? If they are different are there standard names to refer to them (like transformation matrix and data matrix) or should we infer it from the context.
Thanks.
 A: An $(m\times n)$-matrix is a rectangular array of "things of the same type", like numbers, colors, group elements, prices, etc., whereby this array has $m$ lines and $n$ columns. Matrices occur in many different parts of mathematics and daily life, and are used for very different purposes. Even in the field of linear algebra matrices are used at different places: as matrices of linear transformations, of coordinate transformations, of systems of linear equations, what have you. When you see a $(4\times3)$-matrix of prices for shirts from four different fabrics and having three different arm lengths you should not ask: Where is the linear transformation?
Depending on the particular purpose a certain matrix has in your context you can very well call it a transformation matrix, or a multiplication table, etc.
A: Two points:

*

*If we think of a matrix as an operator on row-vectors (rather than on column-vectors) with $T(x) = xA$, then it is natural to think of $A$ as a collection of rows in the same way that it would normally be natural to think of it as a collection of columns. Note that because
$$
[T(x)]^\top = [xA]^\top = A^\top x^\top
$$
(where $A^\top$ denotes a transpose), this alternative way of using $A$ is equivalent to looking that the transformation $x \mapsto A^\top x$.


*Even if we associate $A$ with a linear transformation in the usual way (i.e. think of the map $T(x) = Ax$ on column-vectors), there is a reasonable way to interpret the rows of $A$. In particular: because $T:\Bbb R^{n} \to \Bbb R^m$, we can obtain a linear map from $\Bbb R^n \to \Bbb R$ if we look at a specific coordinate of an output.  In other words, we can think of $T$ as the map define by
$$
T(x) = (T_1(x),\dots,T_m(x))^\top,
$$
where each $T_k: \Bbb R^n \to \Bbb R$ is the linear transformation assocaited with the $k$th row of $A$. In order to think of the rows of $A$ as being "vectors", we would say that the $k$th row is the vector $v$ for which $T_k(x) = v \cdot x$. Without the dot-product, we don't have such a geometric interpretation.
