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.
Best Answer
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.