Vector Spaces and Subspaces
Stub. Vector spaces over a field, subspaces (sums and direct sums), linear combinations and span, linear dependence/independence, bases and dimension, and maximal linearly independent subsets.
The axiomatic starting point shared by both treatments: a vector space over a field, its subspaces (including how two subspaces combine — sum and, in the disjoint case, direct sum), and the two central finiteness questions — does a spanning set exist, and what is the minimal size of one (a basis, whose common cardinality is the dimension). Friedberg–Insel–Spence get here via systems of linear equations as motivation; Axler builds it more directly through span and linear independence.
Outline (TODO — flesh out each)
- Vector spaces — axioms, examples,
- Subspaces — sums of subspaces, direct sums
- Linear combinations and span; systems of linear equations as spanning questions
- Linear dependence and independence
- Bases and dimension
- Maximal linearly independent subsets (infinite-dimensional case)