Vector Space

Dependencies:

  1. Group
  2. Field

A vector space $V$ over a field $F$ is a set with a vector addition operation ($V \times V \mapsto V$) and a scalar multiplication operation ($F \times V \mapsto V$) that satisfies all of these axioms:

  1. $(V, +)$ is a commutative group.
  2. Scalar associativity: $\forall a \in F, \forall b \in F, \forall v \in V, a(bv) = (ab)v$.
  3. Distributivity: $\forall a \in F, \forall b \in F, \forall v \in V, (a+b)v = (av + bv)$.
  4. Distributivity: $\forall a \in F, \forall u \in V, \forall v \in V, a(u+v) = (au + av)$.
  5. Unit scalar: $\forall v \in V, 1v = v$.

The elements of $V$ are called vectors. The elements of $F$ are called scalars.

Two examples of vector spaces are $F$ and ${\mathbf{0}}$ (it is trivial to prove that they are vector spaces).

Dependency for:

  1. Convex function
  2. Linearity of expectation
  3. Expected value of a random variable
  4. Convex set
  5. Convex combination and convex hull
  6. Cone
  7. Linear transformation
  8. Inner product space
  9. Incrementing a linearly independent set
  10. Span
  11. Linearly independent set is not bigger than a span
  12. Negation in vector space
  13. Condition for being a subspace
  14. span(A)=span(B) & |A|=|B| & A is linindep ⟹ B is linindep
  15. Zeros in vector space
  16. Affine independence (incomplete)
  17. Linear independence
  18. Dimension of a set of vectors
  19. Decrementing a span
  20. Basis of a vector space
  21. Basis of F^n
  22. Matrices over a field form a vector space
  23. Row space
  24. Vector matroid

Info:

Transitive dependencies:

  1. Group
  2. Ring
  3. Field