Let U be a finite dimensional vector space over F
Posted: Tue Aug 03, 2021 12:04 pm
Let U be a finite dimensional vector space over F and u #0 be any vector of U. Then there exists some fe U* such that f(u) = 0
Since U is a finite-dimensional vector space, then it has a maximum linearly independent subset, which we shall call, BCU , and this basis generates U. Also, if for every * UEU we had f(u) = 0 then we would see that it would be the null space and U ER but we had said earlier that U is finite-dimensional so it contradicts that, and so there must be at least one such that f(u) +0
Since U is a finite-dimensional vector space, then it has a maximum linearly independent subset, which we shall call, BCU , and this basis generates U. Also, if for every * UEU we had f(u) = 0 then we would see that it would be the null space and U ER but we had said earlier that U is finite-dimensional so it contradicts that, and so there must be at least one such that f(u) +0