Linearly independent set can be expanded into a basis

Dependencies:

  1. Basis of a vector space
  2. Linearly independent set is not bigger than a span
  3. Incrementing a linearly independent set

Let $S = \{v_1, v_2, \ldots, v_k\}$ be a linearly independent subset of a vector space $V$. Let $V$ have a basis of size $n$. Then $k \le n$ and $\exists S' = \{v_{k+1}, \ldots, v_n\} \subset V$ such that $S \cup S'$ is a basis of $V$.

Proof

Let $B$ be a basis of $V$. Since $B$ spans $V$ and $S$ is linearly dependent, $|S| \le |B| \Rightarrow k \le n$.

Define $S_k = S$. $|S_k| = k$.

If $S_i$ doesn't span $V$, $\exists v \in V$ which cannot be expressed as a linear combination of $S_i$. Define $S_{i+1} = S_i \cup \{v\}$. If $S_i$ is linearly independent, $S_{i+1}$ is linearly independent. $|S_{i+1}| = |S_i| + 1$.

Therefore, from $S_k$ we can generate $S_{k+1}$ if $S_k$ doesn't span $V$, from $S_{k+1}$ we can generate $S_{k+2}$ if $S_{k+1}$ doesn't span $V$, and so on. We will either eventually get a set $S_m$ which spans $V$, or $S_i$ doesn't span $V$ for all $i \ge k$.

Using mathematical induction, it can be proved that for all $i \ge k$:

Case 1: $S_i$ doesn't span $V$ for all $i \ge k$

Since $S_{n+1}$ is linearly independent and $B$ spans $V$, $|S_{n+1}| \le |B| \Rightarrow n+1 \le n \Rightarrow \bot$.

Therefore, such a case cannot occur.

Case 2: There is a set $S_m$ which spans $V$

This makes $S_m$ a basis of $V$.

Since $S_m$ spans $V$ and $B$ is linearly independent, $|B| \le |S_m|$. Since $S_m$ is linearly independent and $B$ spans $V$, $|S_m| \le |B|$. Therefore, $|S_m| = m = |B| = n$.

Therefore, it is possible to extend $S$ to get a basis of $V$ of size $n$.

Dependency for:

  1. Basis of range of linear transformation
  2. Symmetric operator on V has a basis of orthonormal eigenvectors

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. Group
  5. Ring
  6. Polynomial
  7. Integral Domain
  8. Comparing coefficients of a polynomial with disjoint variables
  9. Field
  10. Vector Space
  11. Linear independence
  12. Span
  13. Incrementing a linearly independent set
  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. Basis of a vector space
  35. Linearly independent set is not bigger than a span
  36. Homogeneous linear equations with more variables than equations
  37. Rank of a homogenous system of linear equations
  38. Rank of a matrix