Problem 4. Recall that in the notes, we thought about the following transformation: Let V be an n-dimensional vector space and let S :={v1, 02, . . . , Tn} be any basis for V. Let EV, and suppose that...

Extracted text: Problem 4. Recall that in the notes, we thought about the following transformation: Let V be an n-dimensional vector space and let S :={v1, 02, . . . , Tn} be any basis for V. Let EV, and suppose that ở = c101 + c22 + + + Cn Un. So define T: V → R" by %3D T(T) := (c1, c2, ..., Cn). (C1, C2, .. . (a) Prove that T is well defined; that is, prove that each vector in V has exactly one image under T. This means proving two things: (1) that every vector in V has an image under T, and (2) that each vector in V has only one image under T. (b) Prove that T is a linear transformation. (c) Prove that T is one to one. (d) Prove that T is onto. 1