Assume T:Rn to Rm is a linear map. Let Vi be an element in Rn, i=1,..,k. Prove or disprove: if {TV1,....,TVk} is LI then {Vi,...,Vk} is L.I. Solution TV1,TV2,.......TVk are linearly independent.hence c1TV1+c2TV2+................+ckTVk = 0 iff c1,c2,.....ck=0 since T is linear map .hence T(c1V1)+T(c2V2)+.................+T(ckVk)=0 iff c1,c2,.....ck=0 T(c1V1+c2V2+...........ckVk) = 0 iff c1,c2,.....ck=0 T(c1V1+c2V2+...........ckVk) = 0 if c1V1+c2V2+...........ckVk = 0 hence c1V1+c2V2+...........ckVk = 0 iff c1,c2,.....ck=0 so proved that V1,V2,................Vk are linearly independent..

1. Assume T:Rn to Rm is a linear map. Let Vi be an element in Rn, i=1,..,k.
Prove or disprove: if {TV1,....,TVk} is LI then {Vi,...,Vk} is L.I.
Solution
TV1,TV2,.......TVk are linearly independent.hence
c1TV1+c2TV2+................+ckTVk = 0 iff c1,c2,.....ck=0
since T is linear map .hence
T(c1V1)+T(c2V2)+.................+T(ckVk)=0 iff c1,c2,.....ck=0
T(c1V1+c2V2+...........ckVk) = 0 iff c1,c2,.....ck=0
T(c1V1+c2V2+...........ckVk) = 0 if c1V1+c2V2+...........ckVk = 0
hence
c1V1+c2V2+...........ckVk = 0 iff c1,c2,.....ck=0
so proved that V1,V2,................Vk are linearly independent.