Since m <= n, the only choices are m < n and m = n. If m = n, then from
we have
i.e., L is not linearly independent, contrary to the assumption.
Hello everyone, im having trouble understanding the Replacement Theorem proof outline in Friedberg Linear Algebra 4th pg 45.
Theorem 1.10 (Replacement Theorem)
Let be a vector space that is generated by a set containing exactly vectors, and let be a linearly independent subset of containing exactly vectors. Then and there exists a subset of containing exactly vectors such that generates
The proof is by induction on , I skipped the base case .
Inductive step
Let , be a linearly independent subset of consisting of vectors. Then is linearly independent, and so we may apply the induction hypothesis to conclude that and that theres is a subset of such that generates . Thus there exist scalars such that .
Note that , lest be a linear combination of which contradicts the assumption that L is linearly independent.
I am having trouble understanding why does .
Could someone give me a hint so I can understand. Thanks!
Edit: Is it beacause the induction hypothesis gives us and and since has an extra vector means that they cannot be equal?