Linearly independent set is not bigger than a span

Dependencies:

  1. Vector Space
  2. Linear independence
  3. Span
  4. Matrix
  5. Homogeneous linear equations with more variables than equations

Let $U = [u_1, u_2, \ldots, u_m]$ and $V = [v_1, v_2, \ldots, v_n]$ be sequences of vectors from a vector space over field $F$. If $U \subseteq \operatorname{span}(V)$ and $m > n$, then $U$ is linearly dependent.

Proof

Since $U \subseteq \operatorname{span}(V)$, let $u_i = \sum_{j=1}^n a_{i, j}v_j$. Let $A$ be an $m$-by-$n$ matrix over field $F$ where $A_{i, j} = a_{i, j}$.

Consider the system of equations $A^Tx = 0$. This is a system of $n$ equations in $m$ variables. Since $n < m$, there exists a non-zero solution $b$, i.e., $b \in F^m - \{0\}$ such that $A^Tb = 0$. Hence, \[ \sum_{i=1}^m b_iu_i = \sum_{i=1}^m \sum_{j=1}^n b_ia_{i,j}v_j = \sum_{j=1}^n (A^Tb)_jv_j = 0. \] Therefore, $U$ is linearly dependent.

Dependency for:

  1. A matrix is full-rank iff its rows are linearly independent
  2. span(A)=span(B) & |A|=|B| & A is linindep ⟹ B is linindep
  3. Dimension of a set of vectors
  4. Basis of a vector space
  5. Minimally spanning iff basis
  6. Linearly independent set can be expanded into a basis
  7. Maximally linearly independent iff basis
  8. Spanning set of size dim(V) is a basis
  9. A set of dim(V) linearly independent vectors is a basis
  10. Vector matroid
  11. Dimension of a polyhedron

Info:

Transitive dependencies:

  1. /linear-algebra/matrices/gauss-jordan-algo
  2. /linear-algebra/vector-spaces/condition-for-subspace
  3. /sets-and-relations/equivalence-relation
  4. Group
  5. Ring
  6. Polynomial
  7. Field
  8. Vector Space
  9. Linear independence
  10. Span
  11. Integral Domain
  12. Comparing coefficients of a polynomial with disjoint variables
  13. Semiring
  14. Matrix
  15. Stacking
  16. System of linear equations
  17. Product of stacked matrices
  18. Matrix multiplication is associative
  19. Reduced Row Echelon Form (RREF)
  20. Elementary row operation
  21. Every elementary row operation has a unique inverse
  22. Row equivalence of matrices
  23. Matrices over a field form a vector space
  24. Row space
  25. Row equivalent matrices have the same row space
  26. RREF is unique
  27. Identity matrix
  28. Inverse of a matrix
  29. Inverse of product
  30. Elementary row operation is matrix pre-multiplication
  31. Row equivalence matrix
  32. Equations with row equivalent matrices have the same solution set
  33. Homogeneous linear equations with more variables than equations
  34. Rank of a homogenous system of linear equations
  35. Rank of a matrix
  36. Basis of a vector space