Prove the following: 1. Let V and W be vector spaces. Let {v(1), v(2), ..., v(p)

Question

Prove the following: 1. Let V and W be vector spaces. Let {v(1), v(2), ..., v(p)} be a basis of V and let {w(1), w(2), ..., w(p)} be an arbitrary set of vectors. Show that there is a unique linear transformation such that T(v(i)) = w(i) for all i. 2. Let V and W be vector spaces. Let {v(1), v(2), ..., v(p) be a basis of V and let {w(1), w(2), ..., w(p)} be a basis of W. Show that a linear transformation T : V => W such that T(v(i)) = w(i) for all i is an isomorphism. 3. Let T be an isomorphism between vector spaces V and W. Show that {v(1), v(2), ..., v(p)} is a basis if and only if {T(v(1)), T(v(2)), ... T(v(p))} is a basis.

Explanation / Answer

ANSWER IS AT www.math.uri.edu/~eaton/NotesCh2.pdf

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