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SUMS AND DIRECT SUMS OF VECTOR SUBSPACES

Sum of two subspaces. Let U and W be subspaces of a vector space V. The sum of U and W, written U + W, consists of all sums u + w where u є U and w є W.

Example. Pass any plane through the origin of an x-y-z Cartesian coordinate system. Denote the plane by K. Plane K is a subspace of three dimensional space. Designate it as vector space V. Let A and B be any two non-collinear lines passing through the origin and lying in plane K. Each line represents a subspace of plane K. Designate line A as vector space U and line B as vector space W. Then V = U + W.

Theorem 1. The sum U + W of the subspaces U and W of V is also a subspace of V.

Direct sum of vector subspaces. Let M1, M2, .... , Mn ( n 2 ) be subspaces in a vector space V. We say that this set of subspaces is linearly independent if no Mi contains a nonzero vector which is in the subspace determined by the remaining n-1 subspaces. The subspace generated by the elements of M1 ∪ M2 ∪ .... ∪ Mn is denoted by M1 ⊕ M2 ⊕ .... ⊕ Mn and called the direct sum of M1, M2, .... , Mn . Elements x of the direct sum are representable uniquely in the form

Example. Let M1, M2, M3 represent three linearly independent vectors of three dimensional Euclidean space. The direct sum of M1, M2, and M3 is the entire three dimensional space. Any vector x in three dimensional space can be represented as

Theorem 2. The vector space V is the direct sum of its subspaces U and W if and only if :

1. V = U + W

2. U W = {0}             (i.e. U and W are disjoint)

Theorem 3. If the finite-dimensional vector space V is the direct sum of its subspaces S and T, then the union of any basis of S with any basis of T is a basis of V.

Theorem 4. If the finite-dimensional vector space V is the direct sum of its subspaces S and T, then

dim V = dim S + dim T.

(i.e. the dimension of V is equal to the dimension of S plus the dimension of T).

Def. Complementary subspaces. When V is the direct product of S and T, then we call S and T complementary subspaces of V.

Theorem 5. Let S and T be any two finite-dimensional subspaces of a vector space V. Then

dim S + dim T = dim (S T) + dim (S+T)