In the text i am referring for Linear Algebra , following definition for Infinite dimensional vector space is given .
The Vector Space V(F) is said to be infinite dimensional vector space or
infinitely generated if there exists an infinite subset S of V such that L(S) = V.
I am having following questions which the definition fails to answer :-
- Let us say i have an infinite dimensional vector space I(F) .If S is the infinite subset that spans I(F) , can i say that S is Linearly Independent . If so , why ?
- Does that mean any subset that spans every infinite dimensional vector space is Linearly Independent.
- On similar lines , can i say that Basis will exist for each Infinite Dimensional Vector Space and it is nothing but the subset that spans over vector space ?
the
carries some connotation in English that you think it is the only possible object that fits that description. Vector spaces in general can have many different spanning sets. You should have used the articlea
oran
here instead implying you are aware that there could be others. Now... a spanning set does not need to be linearly independent. A basis does. Don't confuse these. As for your supposed definition you quoted, it is missing the condition that $S$ be linearly independent. $\endgroup$