An internal direct sum is simply a direct sum of subobjects of an object.
For example, the real vector space R2 = {(x, y) : x, y ∈ R} is the direct sum of the x-axis {(x, 0) : x ∈ R} and the y-axis {(0, y) : y ∈ R}, and the sum of (x, 0) and (0, y) is the "internal" sum in the vector space R2; thus, this is an internal direct sum. More generally, given a vector space V and two subspaces U and W, V is the (internal) direct sum U ⊕ W if
- U + W = {u + w : u ∈ U, w ∈ W} = V, and
- if u + w = 0 with u ∈ U and w ∈ W, then u = w = 0.
In other words, every element of V can be written uniquely as the sum of an element in U with an element of W
Another case is that of abelian groups. For example, the Klein four-group V = {e, a, b, ab} is the (internal) direct sum of the cyclic subgroups <a> and <b>.
By contrast, a direct sum of two objects which are not subobjects of a common object is an external direct sum. Note however that "external direct sum" is also used to refer to an infinite direct sum of groups, to contrast with the (larger) direct product.
Read more about this topic: Direct Sum
Famous quotes containing the words internal, direct and/or sum:
“I maintain that I have been a Negro three times—a Negro baby, a Negro girl and a Negro woman. Still, if you have received no clear cut impression of what the Negro in America is like, then you are in the same place with me. There is no The Negro here. Our lives are so diversified, internal attitudes so varied, appearances and capabilities so different, that there is no possible classification so catholic that it will cover us all, except My people! My people!”
—Zora Neale Hurston (1891–1960)
“The most passionate, consistent, extreme and implacable enemy of the Enlightenment and ... all forms of rationalism ... was Johann Georg Hamann. His influence, direct and indirect, upon the romantic revolt against universalism and scientific method ... was considerable and perhaps crucial.”
—Isaiah Berlin (b. 1909)
“To die proudly when it is no longer possible to live proudly. Death freely chosen, death at the right time, brightly and cheerfully accomplished amid children and witnesses: then a real farewell is still possible, as the one who is taking leave is still there; also a real estimate of what one has wished, drawing the sum of one’s life—all in opposition to the wretched and revolting comedy that Christianity has made of the hour of death.”
—Friedrich Nietzsche (1844–1900)