A matrix is full-rank iff its rows are linearly independent

Dependencies:

  1. Rank of a matrix
  2. Row equivalent matrices have the same row space
  3. Rows of RREF are linearly independent
  4. Basis of a vector space
  5. Linearly independent set is not bigger than a span
  6. Spanning set of size dim(V) is a basis

Let $A$ be an $m$ by $n$ matrix. Then $\operatorname{rank}(A) = m \iff$ rows of $A$ are linearly independent.

Proof

Let $B = \operatorname{RREF}(A)$. Therefore, $B$ is row-equivalent to $A$. Therefore, $A$ and $B$ have the same row-space. Let that row-space be $V$.

Let $S$ be the set of non-zero rows in $B$. Let $|S| = k$. Since $S$ spans $V$ and $S$ is linearly independent, $S$ is a basis of $V$.

Let $L$ be the set of rows of $A$. Since $L$ spans $V$ and $S$ is a linearly independent subset of $V$, $|S| \le |L|$.

Proof of 'only-if' part

$L$ is a linearly independent subset of $V$ and $S$ spans $V$. Therefore, $|L| \le |S| \Rightarrow |L| = |S|$. Therefore, $\operatorname{rank}(A) = m$.

Proof of 'if' part

Let $\operatorname{rank}(A) = m$. Therefore, $|S| = |L|$.

Since $L$ is a spanning set of $V$ of size $|S|$ and $S$ is a basis of $V$, $L$ is also a basis of $V$. Therefore, $L$ is linearly independent.

Dependency for:

  1. A is diagonalizable iff there are n linearly independent 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. Decrementing a 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. Rows of RREF are linearly independent
  22. Matrices over a field form a vector space
  23. Row space
  24. Elementary row operation
  25. Every elementary row operation has a unique inverse
  26. Row equivalence of matrices
  27. Row equivalent matrices have the same row space
  28. RREF is unique
  29. Identity matrix
  30. Inverse of a matrix
  31. Inverse of product
  32. Elementary row operation is matrix pre-multiplication
  33. Row equivalence matrix
  34. Equations with row equivalent matrices have the same solution set
  35. Rank of a matrix
  36. Basis of a vector space
  37. Linearly independent set is not bigger than a span
  38. Homogeneous linear equations with more variables than equations
  39. Rank of a homogenous system of linear equations
  40. Spanning set of size dim(V) is a basis