New vector spaces from old
The main examples are the following:
Given a vector space X and a subset Y of X. Then Y is called a subspace of X if it's a vector space. The inclusion iY: Y → X, iY(x) = x, is a linear operator.
Given two vector spaces X and Y. Then X×Y is the product of sets, set of all pairs (a,b) of elements in X and Y respectively. It is a vector space with the operations (a,b) + (a',b') = (a + a',b + b') and t(a,b) = (ta,tb). The projections pX: X×Y → X and pY: X×Y → Y, pX(a,b) = a, pY(a,b) = b, are linear operators.
Given a vector space X and a subspace Y. Then an equivalence relation ~ on X is defined by x ~ y if x - y ∈ Y. Then the quotient set X/~ is a vector space with the operation [x] + [y] = [x + y] and q[x] = [qx]. The quotient function q: X → X/~, q(x) = [x], is a linear operator.