Mark the statement either true (in all cases) or false (for at least one example). if false, construct a specific example to show that the statement is not always true. such an example is called a counterexample to the statement. if the statement is true, give a justification. if v1,…,v4 are in r4 and {v1,v2,v3} is linearly dependent then {v1,v2,v3,v4} is also linearly dependent.
select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
(a) true. because v3 = 2v1 + v2, v4 must be the zero vector. thus, the set of vectors is linearly dependent.
(b) true. the vector v3 is a linear combination of v1 and v2, 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 c1 =2, c2 = 1, c3 = 1, and c4 = 0, then c1v1 + + c4v4 =0. the set of vectors is linearly dependent.
(d) false. if v1 = v2 = v3 = and v4 = [1 2 1 2], then v3 = 2v1 + v2 and {v1, v2, v3, v4} is linearly independent.