span(A)=span(B) & |A|=|B| & A is linindep ⟹ B is linindep

Dependencies:

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

$\newcommand{Span}{\operatorname{span}}$ Let $A$ and $B$ be sets of vectors such that $\Span(A) = \Span(B)$ and $|A| = |B|$ and $A$ is linearly independent. Then $B$ is also linearly independent.

Proof

Let $V = \Span(A) = \Span(B)$. Suppose $B$ is linearly dependent. Then $\exists v \in B$ such that $\Span(B) = \Span(B - \{v\})$. Since $A$ is linearly independent, $A \subseteq V$, and $B - \{v\}$ spans $V$, we get $|A| \le |B - \{v\}| = |B|-1$, which is a contradiction. Hence, $B$ is linearly independent.

Dependency for: None

Info:

• Depth: 6
• Number of transitive dependencies: 38

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. 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