[Linear Algebra; Logic] I'm trying to rewrite the statement of a theorem and I'm not sure if I have done it correctly.
Theorem: In a finite-dimensional vector space, the length of every linearly independent list of vectors is less than or equal to the length of every spanning list of vectors.
My version: If V is a finite-dimensional vector space, then for all lists t of vectors in V and for every spanning list p in V, if t is linearly independent, then len(t) ≤ len(p) (where len(x) is the length of x).
However, if you take the contrapositive, then you have: if len(t) > len(p), then t is linearly dependent. But since the quantifier is not negated as well, this is saying that the length of t is greater than the length of every spanning list, which I don't think makes sense. Did I go wrong somewhere?
Posted Jul 26, 2014, 9:27 PM UTC
15
Contrapositive of "A implies B" is "not B implies not A". Here "If p is a spanning list, and len(t)>len(p) then t is not linearly independent." which is equivalent and true.
Didn't you change the quantifier on p from universal to existential, though?
The statement I formed the contrapositive of is: "t linearly independent implies len(t)<=len(p)". There are no quantifiers in that statement.
Right, but if p is any spanning list, then does it make sense to say that some list can be longer than any spanning list?
Can you make that a formal statement? Then, what statement are you starting with and how does it logically imply your conclusion?