Affine independence (incomplete)

Dependencies:

1. Vector Space
2. Linear independence

$$ Let $X=\{x^{(1)},\dots ,x^{(n)}\}$ be vectors over field $F$. $X$ is affinely independent iff (the following conditions are equivalent):

1. $\mathrm{\forall }\alpha \in F^n$, if $\sum _{i=1}^n{\alpha }_i=0$ and $\sum _{i=1}^n{\alpha }_i{x}^{(i)}=0$, then $\alpha =0$.
2. $\{x^{(1)}-x^{(n)},\dots ,x^{(n-1)}-x^{(n)}\}$ is linearly independent.

Proof of equivalence (incomplete)

Dependency for:

1. Dimension of a polyhedron
2. Dimension of a set of vectors

Info:

• Depth: 5
• Number of transitive dependencies: 5

Transitive dependencies:

1. Group
2. Ring
3. Field
4. Vector Space
5. Linear independence