What happens geometrically when one multiplies a matrix A by the inverse of a matrix B?
Consider the rows of B as a basis b_1, … b_n, and the rows of A as vectors (not necessarily a basis) a_1, .. , a_n. Then AB^{-1} is the unique linear map that sends
b_i --> a_i for all i =1, … , n.
Not exactly lower-school stuff, perhaps secondary school stuff. It is never wrong to wonder at a technical operation (matrix multiplication) and realizing that it performs the details of a fairly abstract operation of composing two (linear) functions. The second matrix has to be inverted first, which by itself is a technical performance of reversing a (reversible) function.
If the first (right-hand) matrix A represents the function f and the other matrix B (to be inverted still) represents the function g, the matrix product B-1*A represents the composed function "first f, then g-1", denoted g-1 o f (where the symbol "o" is composition). It takes a vector x to f(x), which is seen as g(y) for some vector y, and then f(x) alias g(y) is finally taken to y. The computed matrix will transform x to this y.
Peter: My mistake. I was taken away by the fact that the Dutch (rather, Flemish) terminology for "primary school" ("lagere school") is a literal translation of the English expression "lower school".
Consider the rows of B as a basis b_1, … b_n, and the rows of A as vectors (not necessarily a basis) a_1, .. , a_n. Then AB^{-1} is the unique linear map that sends
b_i --> a_i for all i =1, … , n.
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