Abstract linear algebra

Question

generalize the notions of external and internal direct sums to three or more summands.

Answer 1

External direct sum of three vector spaces U, W and X: U ⊕ W ⊕ X is the set of all 3-tuples of the form (u, w, x)^T where u ∈ U, w ∈ W and x ∈ X. The superscript T is transpose- i.e., I'm thinking of (u, w, x)^T as a column. There's a subtlety: I could form (U ⊕ W) ⊕ X- an element of which would look like ((u, w), x)^T. Or I could form U ⊕ (W ⊕ X)- an element of which would look like (u, (w, x))^T. But not to worry, as the two are isomorphic. For the external direct sum of n > 3 vector spaces, it's the same idea.

Internal direct sum: let U, W and X be subspaces of a vector space V. If dim(U) + dim(W) + dim(X) = dim (V) and the pairwise intersections U intersect W, W intersect X, and U intersect X are all { 0 } (i.e., only the zero vector), then V is equal to the internal direct sum U ⊕ W ⊕ X. To generalize to n > 3 subspaces, you must have all of the pairwise intersections among the subspaces equal to { 0 }.