Determine if the following statements are TRUE or FALSE. Justify your answers.\ (a) If

(T(v_(1)),dots,T(v_(n)))

is linearly independent for some linear map

T

, then

(v_(1),dots,v_(n))

is\ linearly independent.\ (b) If

(T(v_(1)),dots,T(v_(n)))

is linearly independent for some

TinL(V,W)

and basis

(v_(1),dots,v_(n))

\ of

V

, then

T

is

1-1

.\ (c) If

(v_(1),dots,v_(n))

is a basis for

V

and

TinL(V,W)

, then

(T(v_(1)),dots,T(v_(n)))

is a basis for\ range

(T)

.\ (d)

span(v_(1),dots,v_(n))=V

if and only if for every non-zero

linV^(')

, there is some

k

for which\

l(v_(k))!=0

.

1. Determine if the following statements are TRUE or FALSE. Justify your answers. (a) If (T(v1),,T(vn)) is linearly independent for some linear map T, then (v1,,vn) is linearly independent. (b) If (T(v1),,T(vn)) is linearly independent for some TL(V,W) and basis (v1,,vn) of V, then T is 11. (c) If (v1,,vn) is a basis for V and TL(V,W), then (T(v1),,T(vn)) is a basis for range (T). (d) span(v1,,vn)=V if and only if for every non-zero V, there is some k for which (vk)=0