Let t: set of real numbers r^n→set of real numbers r^m be a linear transformation, and let {v1, v2, v3} be a linearly dependent set in set of real numbers r^n.
explain why the set {t(v1), t(v2), t(v3)} is linearly dependent. choose the correct answer below.
a. given that the set {v1, v2, v3} is linearly dependent, there exist c1, c2, c3, not all zero, such that c1v1+c2v2+c3v3=0. it follows that c1t(v1)+c2t(v2)+c3t(v3)=0. therefore, the set t(v1), t(v2), t(v3)} is linearly dependent.
b. given that the set {v1, v2, v3} is linearly dependent, there exist c1, c2, c3, not all zero, such that c1v1+c2v2+c3v3≠0. it follows that c1t(v1)+c2t(v2)+c3t(v3)≠0. therefore, the set t(v1), t(v2), t(v3)} is linearly dependent.
c. given that the set {v1, v2, v3} is linearly dependent, there exist c1, c2, c3, not all zero, such that c1v1+c2v2+c3v3=0. it follows that c1t(v1)+c2t(v2)+c3t(v3)≠0. therefore, the set t(v1), t(v2), t(v3)} is linearly dependent.
d. given that the set {v1, v2, v3} is linearly dependent, there exist c1, c2, c3, all zero, such that c1v1+c2v2+c3v3=0. it follows that c1t(v1)+c2t(v2)+c3t(v3)=0. therefore, the set t(v1), t(v2), t(v3)} is linearly dependent.