Unformatted text preview: W (the target space) is ﬁnitedimensional used? Why can this assumption be removed (how can the proof be modiﬁed to not need it)? Problem 4. Give an example of a linear transformation T : V → V such that T is injective (oneone) but not surjective (onto). “Hint”: V must be inﬁnitedimensional. Problem 5. Let V and W be vector spaces. Let S ⊆ V be linearly independent but n ot a basis. Let T ⊆ W such that # T = # S . Enumerate S = { s 1 ,...,s n } and T = { t 1 ,...,t n } . Show that there exist at least two distinct linear transformations U 1 : V → W and U 2 : V → W such that U 1 ( s j ) = t j and U 2 ( s j ) = t j for each j = 1 ,...,n . Now let S ⊆ V be a linearly dependent set and T as above. Show that there need not be any linear transformation U such that U ( S ) = T . 1...
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This note was uploaded on 03/15/2012 for the course MATH 115A 262398211 taught by Professor Fuckhead during the Winter '10 term at UCLA.
 Winter '10
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