Two subspaces, two ways to combine them
Recall a subspace is a subset that is itself a vector space (closed under addition and scaling, and containing 0). Given two subspaces U and W of V, two new subspaces appear for free: their intersection U ∩ W (everything in both) and their sum U + W = { u + w : u in U, w in W }, the smallest subspace containing both.
Order them, and you get a lattice
Order subspaces by inclusion: U <= W means U is contained in W. Under this order the whole collection of subspaces becomes a [[subspace-lattice|lattice]]: any two have a greatest lower bound (the meet, = intersection) and a least upper bound (the join, = sum). The bottom of the lattice is {0}; the top is V itself.
A slice of the subspace lattice of R^3 (lines L, planes P):
R^3 <- top (V)
/ | \
P1 P2 P3 <- planes (dim 2)
/ \ / \ / \
L1 L2 L3 ... <- lines (dim 1)
\ | /
{0} <- bottom (dim 0)
meet P_i ^ P_j = a line (planes intersect in a line)
join L_i v L_j = a plane (two lines span a plane)The dimension formula ties it together
The lattice has a beautiful numerical law relating meet and join — the inclusion-exclusion of linear algebra:
dim(U + W) = dim(U) + dim(W) - dim(U ^ W)
Example in R^4:
U = span{ e1, e2, e3 } dim 3
W = span{ e3, e4 } dim 2
U ^ W = span{ e3 } dim 1
U + W = span{e1,e2,e3,e4}=R^4, dim 4
check: 3 + 2 - 1 = 4. OK.Stare at that formula: when the overlap dim(U ∩ W) is zero, the sum's dimension is exactly dim(U) + dim(W) — nothing is double-counted. That clean, lossless case is so important it gets its own name in the next guide: the direct sum.