Span

Dependencies:

1. Vector Space

Let $S = \{v_1, v_2, \ldots, v_n\}$ be a set of $n$ vectors from the vector space $V$ over field $F$.

\[ \operatorname{span}(S) = \left\{ \sum_{i=1}^n a_iv_i : \forall (a_1, a_2, \ldots, a_n) \in F^n \right\} \]

If $W = \operatorname{span}(S)$, we say that $S$ spans $W$.

If $S$ is of infinite size, $\operatorname{span}(S)$ is the set of linear combinations of all finite subsets of $S$.

Dependency for:

1. Gram-Schmidt Process
2. Joining orthogonal linindep sets
3. Linearly independent set is not bigger than a span
4. span(A)=span(B) & |A|=|B| & A is linindep ⟹ B is linindep
5. Decrementing a span
6. Basis of a vector space
7. Vector matroid

Info:

• Depth: 4
• Number of transitive dependencies: 4

Transitive dependencies:

1. Group
2. Ring
3. Field
4. Vector Space