This chapter expands on the notion of vector spaces by introducing the concept and properties of subspaces. It defines subspaces, sums of subspaces, and direct sums.
The motivation for subspaces is kind of similar to vector spaces. Just like vector spaces cannot exist outside the context of a field, subspaces cannot exist outside the context of a vector space(and thus the field itself).
Really, a subspace is sort of similar to a subset. In fact, a subspace U of a vector space V must consist of a subset of V following certain rules: namely, it must follow the rules that dictate that U is also a vector space.
Important Definitions A subset U of V is called a subspace of V if U is also a vector space with the same additive identity, addition, and scalar multiplication as on V.
We can calculate the sum of subspaces, and they’re in some way similar to the union of sets: Suppose \({latex.inline[V_{1}, ..., V_{m}](V_{1}, ..., V_{m})} are subspaces of V. The **sum** of the spaces is the set of all possible sums of elements of \){latex.inlineV{1}, ..., V{m}}. In pure math notation we get: ${latex.inlineV{1} + ... +\ V{n} = {v{1} + ... + v{n} } : v{1} \in V{1},\ ...,\ v{m} \in V{m}}
A sum is a direct sum if each element of the sum can be written in only one way as a sum ${latex.inlinev{1} + ...\ + v{m}}.
Relevant Theorems * 1753142531 - Axler 1.34 Conditions for a subspace * 1753142314 - Axler 1.40 Sum of subspaces is smallest containing subspace * 1753142362 - Axler 1.45 Conditions for a direct sum * 1753142442 - Axler 1.46 Direct sum of two subspaces