(Requires the subsection on Combining Subspaces, which is optional.) Let U and W be vector spaces. Define a new vector space, consisting of the set U × W = {(, ) | ˆˆ U and ˆˆ W} along with these operations.
(1, 1) + (2, 2) = (1 + 2, 1 + 2) and r ˆ™ (, ) = (r, r)
This is a vector space, the external direct sum of U and W.
(a) Check that it is a vector space.
(b) Find a basis for, and the dimension of, the external direct sum P2 × R2.
(c) What is the relationship among dim(U), dim(W), and dim(U × W)?
(d) Suppose that U and W are subspaces of a vector space V such that V = UŠ•W (in this case we say that V is the internal direct sum of U and W). Show that the map f: U × W †’ V given by

is an isomorphism. Thus if the internal direct sum is defined then the internal and external direct sums are isomorphic.

(, w) +w