Let n N and let PnK be the vector space of all polynomials o

Let n N and let P_n(K) be the vector space of all polynomials of degree at most n with coefficients in K. Let c K and let p_0, p_1, ..., p_n P_n(K) be polynomials with p_j(c) = 0 for all 0 lessthanorequalto j lessthanorequalto n. Prove that p_0, p_1, ..., p_n are linearly dependent. Let V be a vector space over K. Prove that V is infinite dimensional iff V has a countably infinite linearly independent subset S = {upsilon_1, upsilon_2, upsilon_3, ...}.

Solution

Solution :- (2) Let V be a vector space over K. V is infinite dimensional.

Suppose V that is infinte dimensional and let (v1, . . . , vn) be a list of linearly independent vectors.
Such a list exits for some n (at least for n = 1 ,since V {0}).
This list cannot span V .
Therefore there exists a vector v_n+1 span(v1, v2, . . . , vn) in V .
But now the expanded list (v1, v2, . . . , v_n, v_n+1) is a list of linearly independent vectors in V .
Since any finite list will not span V , it can be expanded further.
Thus we created a sequence of vectors v1, v2, . . . such that (v1, v2, . . . , vn)
is linearly independent for every positive integer n.

Conversely,

suppose such a sequence exists. Since any spanning list of vectors is
longer thatn any linearly independent list, it follows that any spanning list has
length greater than n for any positive integer n.
Hence, no finite list of vectors will span V and V is infinite dimensional.