Direct Sums of Subspaces

Abstract

If A and B are non-empty subsets of a vector space V over a field F then the subspace spanned by A ∪ B, i.e. the smallest subspace of V that contains both A and B, is the set of linear combinations of elements of A ∪ B. In other words, it is the set of elements of the form

$$ [\sum\limits_{i = 1}^m {{\lambda _i}} {a_i} + \sum\limits_{j = 1}^n {{\mu _j}} {b_j} $$

where a _i ∈ A, b _j ∈ B, and λ _i , µ _j ∈ F. In the particular case where A and B are subspaces of V we have $\sum\limits_{i = 1}^m {{\lambda _i}} {a_i} \in A$ and $\sum\limits_{j = 1}^n {{\mu _j}} {b_j} \in B$, so this set can be described as

$$ \{ a + b;a \in A,b \in B\} . $$