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 .
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?