Matthew S. answered • 02/21/20

PhD in Mathematics with extensive experience teaching Linear Algebra

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**}.