Let T: V-> W be a bijective linear transformation
Prove that if {v_{1},v_{2},...,v_{n}} is a basis for V,
then {T(v_{1}),T(v_{2}),...,T(v_{n})} is a basis for W
Assuming is a basis for V.
To show the set is a basis. We need to show 2 tihings
1)Every vector in W can be written as a linear combination of this set (or this set spans W)
2)This set is linearly independent
The first one is pretty straightforward, since T is bijective it means that for every we can find a coordinate in such that
so that equals . There we picked any arbitray vector in W and wrote it as the linear combination of
The second one is also pretty straight forward. if was linearly dependent, it would have atleast one non zero solution to the homogenous equation, so there is a coordinate such that and Now Pick another non zero vector now observe that and but how can 2 different vectors in go to the same vector in W? it was supposed to be one-to-one, thus only the zero solution must exist for the homogenous equation, thus is linearly independent and thus a basis for W.