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3. Let V be a vector space of finite dimension n > 0 over a field F. Let S be a (possibly infinite) generating set of V. Using the three facts:

(a) Any finite generating set for V contains at least n vectors;

(b) Every basis of V contains the same number of vectors;

(c) Every linearly independent subset of V can be extended to a basis for V ,

Prove the following three assertions

. (1) S contains a nonzero vector (10 points).

. (2) If a subset T of S is linearly independent, then the number of elements of T does not exceed n (15 points).

. (3) If T is a linearly independent subset of S with less than n element, we can find a larger linearly independent subset T′ of S containing T (20 points).

4.let v be the Z2-vector space of all function from the set {a,b,c} of three letters into Z2 (here Z2 is the 2 element field made up of 0,1).

a)What is the zero vector in V? specify its value at a, b, c

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