Vector matroid

Dependencies:

  1. Matroid
  2. Vector Space
  3. Linear independence
  4. Span
  5. Linearly independent set is not bigger than a span

Let $V$ be a vector space. Let $S = \{v_1, v_2, \ldots, v_n\} \subseteq V$. Let $I = \{X \subseteq S: X \textrm{ is linearly independent} \}$. Then $(S, I)$ is a matroid, called the vector matroid.

Proof

Let $A \subseteq B$ and $B \in I$.

\begin{align} & B \in I \\ &\Rightarrow B \textrm{ is linearly independent} \\ &\Rightarrow \textrm{Every non-trivial linear combination of } B \textrm{ is non-0} \\ &\Rightarrow \textrm{Every non-trivial linear combination of } A \textrm{ is non-0} \\ &\Rightarrow A \textrm{ is linearly independent} \\ &\Rightarrow A \in I \end{align}

Therefore, $(S, I)$ satisfies the hereditary property.

Suppose $A, B \in I$ and $|A| < |B|$. Assume that $\forall v \in B, A + v$ is linearly dependent. Since $A$ is linearly independent, $v$ is a linear combination of $A$. Therefore, $B \subseteq \operatorname{span}(A)$.

Since $B \subseteq \operatorname{span}(A)$ and $|B| > |A|$, $B$ is linearly dependent, which is a contradiction. Therefore, $\exists v \in B, A + v$ is linearly independent.

Therefore, $(S, I)$ satisfies the exchange property.

Dependency for: None

Info:

Transitive dependencies:

  1. /linear-algebra/vector-spaces/condition-for-subspace
  2. /linear-algebra/matrices/gauss-jordan-algo
  3. /sets-and-relations/equivalence-relation
  4. Matroid
  5. Group
  6. Ring
  7. Polynomial
  8. Integral Domain
  9. Comparing coefficients of a polynomial with disjoint variables
  10. Field
  11. Vector Space
  12. Linear independence
  13. Span
  14. Semiring
  15. Matrix
  16. Stacking
  17. System of linear equations
  18. Product of stacked matrices
  19. Matrix multiplication is associative
  20. Reduced Row Echelon Form (RREF)
  21. Matrices over a field form a vector space
  22. Row space
  23. Elementary row operation
  24. Every elementary row operation has a unique inverse
  25. Row equivalence of matrices
  26. Row equivalent matrices have the same row space
  27. RREF is unique
  28. Identity matrix
  29. Inverse of a matrix
  30. Inverse of product
  31. Elementary row operation is matrix pre-multiplication
  32. Row equivalence matrix
  33. Equations with row equivalent matrices have the same solution set
  34. Linearly independent set is not bigger than a span
  35. Homogeneous linear equations with more variables than equations
  36. Rank of a homogenous system of linear equations
  37. Rank of a matrix
  38. Basis of a vector space