what we would like to do is define .
first we need to be sure that this is well-defined, since we are working with cosets, instead of elements.
so suppose y is in x + X (that is, that y + X = x + X), so that y = x + v, for some vector v in X. then f(y) = f(x + v) = f(x) + f(v) = f(x) + v',
(where v' is some element of X) since X is invariant for f, so f(y) is in f(x) + X, thus f(y) + X = f(x) + X.
hence , so is indeed well-defined.
from here. it's all down-hill, the linearity of is a direct consequence of the linearity of f:
it might be helpful to see a simple concrete example.
let , the Euclidean plane, with the usual vector operations, and suppose . what do the elements of V/X look like?
well anything in (x,y) + X has a 2nd coordinate of y, so the elements of V/X are all horizontal lines (we get one for each different real number y).
so suppose f(x,y) = (3x+y, 2y). it should be clear X is an invariant subspace for f.
then is the mapping that takes the line going through y, to the line going through 2y. in other words, acts "just like" the function a-->2a (of one real variable).
the reason being, when we act "mod X", we are "shrinking" the entire x-dimension down to 0. so what f does on the first coordinate becomes irrelelvant, as far as is concerned.