User:Brithans/Replacement Theorem

The replacement theorem is a theorem in linear algebra that states:

Let V is a vector space and span(E) = V such that E contains exactly n vectors. If F is a linearly independent subset of V containing m vectors, then: 1) m is less than or equal to n    2) G is a subset of E containing exactly n-m vectors such that span(F union G) = V.

Proof of the replacement theorem:

Mathematical induction will be used to prove the replacement theorem: Base case: P(0): If m=0, F = {} (reads, F is a null set), since F has m vectors, and thus G=E holds true.

Induction hypothesis: P(k): Assume an integer k>=0, such that k<=n and G contains n-k vectors and F union G spans V.

Induction step: P(k+1): page 46 "Linear Algebra" Friedberg, Insel, and Spence. Theorem 1.10