Vector Spaces¶
Recall (doc/postdoc). Not a first course.
Job: modules over a field; bases and dimension; linear maps classified by rank–nullity.
Core¶
\(V\) over field \(F\): abelian group \(+\) with scalar mult. \(F\times V\to V\) (distributivity, \(1v=v\), associativity of scalars).
| Concept | Definition |
|---|---|
| Subspace | closed under \(+\) and scalars; \(0\in W\) |
| Span | all finite \(F\)-linear combinations |
| Independent | \(\sum c_iv_i=0\Rightarrow\) all \(c_i=0\) |
| Basis | independent spanning set; \(\dim V=\) cardinality |
| Coordinates | \([v]_B\) coeffs in ordered basis \(B\) |
| Rank–nullity | \(\operatorname{rank}T+\operatorname{nullity}T=\dim V\) for \(T:V\to W\) |
Examples: \(F^n\); \(F[x]_{\le n}\); \(C[0,1]\) (infinite dim); \(\ker T\), \(\operatorname{im}T\).
Isomorphism: finite-dim \(V\cong F^{\dim V}\). Dual \(V^*=L(V,F)\); \((V^*)^*\cong V\) if finite dim.
Used later for¶
- Eigenvalues: invariant subspaces
- Modules: drop field \(\to\) PID classification
- Functional analysis: Banach/Hilbert
- Representation theory: \(G\)-modules over \(F\)
Check¶
Does every linearly independent set extend to a basis?
Yes in finite dim (and in general with Zorn / AC for arbitrary dim). Without choice, pathological issues arise for some spaces.