The statement is either true in all cases or false. If false, construct a specific example to show that the statement is not always true. If v_1, middot, v_4 are in R^4 and v_3 = 2v_1 + v_2, then {v_1, v_2, v_3, v_4} is linearly dependent.
a) True. Because v_3 = 2v_1 +v_2, v_4 must be the zero vector. Thus, the set of vectors is linearly dependent.
b) True. The vector v_3 is a linear combination of v_1 and v_2, so at least one of the vectors in the set is a linear combination of the others and set is linearly dependent.
c) True. If c_1 =2, c_2 = 1, c_3 = 1, and c_4 = 0, then c_1v_1 + middot middot middot + c_4v_4 =0. The set of vectors is linearly dependent.
d) False. If v_1 =, v_2 =, v_3 =, and v_4 = [1 2 1 2], then v_3 = 2v_1 + v_2 and {v_1, v_2, v_3, v_4} is linearly independent.