Let U and W be subspaces of the vector space V. We know that the intersection U∩W is also a subspace of V, whereas the union U∪W is, in general,
not a subspace. In this project you will explore the sum and direct sum of subspaces,
focusing especially on their geometric interpretation in R^n.
1. Define the sum of the subspaces U and W as follows.
U+W= {u+w: u∈U, w∈W}
Prove that U+W is a subspace of V.
2. Consider the subspaces of V= R^3 listed below.
U= {(x, y,x-y): x, y ∈R}
W= {(x,0,x):x ∈R}
Z={(x, x,x): x ∈R}
Find U+W, U+Z, W+Z.
3. If U and W are subspaces of V such that V= U +W and U +W= {0}, prove that every vector in V has a unique representation of the form u+w, where u is in U and wis in W. In this case, we say that V, is the direct sum of U and W and write V=U⊕W.
(Direct sum) Which of the sums in part (2) of this project are direct sums?
4. Let V=U⊕W and suppose that {u1, u2, uk} is a basis for the
subspace U and {w1, w2, wm} is a basis for the subspace W. Prove that the set {u1,uk + w1,wm} is a basis for V.
5. Consider the subspaces of listed below.
U= {(x,0,y): x, y ∈R}
W= {(0,x, y):x, y ∈R}
Show that R^3= U+W. Is R^3 the direct sum of U and W? What are the
dimensions of U, W, U∩W, and U+W? In general, formulate a conjecture that relates the dimensions of U, W, U∩W, and U+W.
6. Can you find two 2-dimensional subspaces of R^3 whose intersection is just
the zero vector? Why or why not?