# Thread: Prove: If two transformations create the same map, they are equal

1. ## Prove: If two transformations create the same map, they are equal

Let
V be an F-vector space with basis B = (v1, v2, . . . , vn). Let W be any
F-vector space and let w1,w2, . . . ,wn be some elements.

Let

S, T : V W be two linear transformations. Suppose S(vi) = T(vi) for i =1, 2, . . . , n. Show that S = T.

2. Originally Posted by amm345
Let
V be an F-vector space with basis B = (v1, v2, . . . , vn). Let W be any
F-vector space and let w1,w2, . . . ,wn be some elements.

Let

S, T : V W be two linear transformations. Suppose S(vi) = T(vi) for i =1, 2, . . . , n. Show that S = T.
HINT: Use linearity of mappings to show that the image of your basis forms a spanning set for the whole image.