Basis of a vector space

Dependencies:

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

Let $B$ be a subset of a vector space $V$. $B$ is a basis of $V$ iff $B$ is linearly independent and $V = \operatorname{span}(B)$.

All bases of a vector space have the same size (either they are all of infinite size or they are all of the same finite size).

If $V$ has a finite basis of size $n$, we say that $\operatorname{dim}(V) = n$. $\operatorname{dim}(V)$ is well-defined since all bases of $V$ have the same size.

Proof that all bases have the same size

Let $B_1$ and $B_2$ be 2 bases of a vector space $V$ and one of them is finite. Without loss of generality, assume $B_1$ is finite.

Assume $B_2$ is larger than $B_1$ ($|B_2| > |B_1|$). Let $S$ be a finite subset of $B_2$ which is larger than $B_1$. Since $B_1$ spans $V$ and $S$ is larger than $B_1$, $S$ is linearly dependent. This means $B_2$ is linearly dependent, which contradicts the fact that $B_2$ is a basis of $V$. Therefore, no subset of $B_2$ is larger than $B_1$. Therefore, $|B_2| \le |B_1|$.

This means $B_2$ is finite. Using a similar argument as above and interchanging the roles of $B_1$ and $B_2$, we can prove that $|B_1| \le |B_2|$. Therefore, $|B_1| = |B_2|$.

Dependency for:

  1. Rank of a matrix
  2. A matrix is full-rank iff its rows are linearly independent
  3. Symmetric operator iff hermitian
  4. Symmetric operator on V has a basis of orthonormal eigenvectors
  5. Dimension of a set of vectors
  6. Preserving a basis by replacing a vector
  7. Coordinatization over a basis
  8. Minimally spanning iff basis
  9. Linearly independent set can be expanded into a basis
  10. Basis changer
  11. Maximally linearly independent iff basis
  12. Spanning set of size dim(V) is a basis
  13. A set of dim(V) linearly independent vectors is a basis
  14. Basis of F^n
  15. Rank of a homogenous system of linear equations
  16. Orthonormal basis change matrix
  17. Standard normal random vector on vector space
  18. Pointing 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. Linearly independent set is not bigger than a span
  34. Homogeneous linear equations with more variables than equations
  35. Rank of a homogenous system of linear equations
  36. Rank of a matrix