Sums and direct sums of vector subspaces

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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, .... , M_n ( n 2 ) be subspaces in a vector space V. We say that this set of subspaces is linearly independent if no M_i 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 ∪ .... ∪ M_n is denoted by M₁ ⊕ M₂ ⊕ .... ⊕ M_n and called the direct sum of M₁, M₂, .... , M_n . 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)