We’ve said before that the category of vector space is enriched over itself. That is, if we have vector spaces and over the field , the set of linear transformations is itself a vector space over . In fact, it inherits this structure from the one on . We define the sum and the scalar product
for linear transformations and from to , and for a constant . Verifying that these are also linear transformations is straightforward.
So what do these structures look like in the language of matrices? If and are finite-dimensional, let’s pick bases of and of . Now we get matrix coefficients and , where indexes the basis of and indexes the basis of . Now we can calculate the matrices of the sum and scalar product above.
We do this, as usual, by calculating the value the transformations take at each basis element. First, the sum:
and now the scalar product:
so we calculate the matrix coefficients of the sum of two linear transformations by adding the corresponding matrix coefficients of each transformation, and the matrix coefficients of the scalar product by multiplying each coefficient by the same scalar.