10w_math115a_hw4 - W (the target space) is nite-dimensional...

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Math 115A Homework # 4 Prof. Yehuda Shalom TA: Darren Creutz Date Due: 4 February 2010 All page numbers and problems numbers refer to “Linear Algebra” by Friedberg, Insel and Spence (4th edition). For questions involving deciding if a statement is true or false, in addition to stating true or false, you must include a brief reason (one line or less: for example name an axiom or cite a theorem). Section 2.1 (p.74-p.79) # 1 (a)-(h), # 2, # 4, # 7, # 13, # 14, # 17, # 19, # 28, # 35. Problem 1. Let V and W be finite-dimensional vector spaces. Let T,U : V W be linear transformations. Prove that N ( T ) N ( U ) N ( T + U ). Is it possible that N ( T + U ) = N ( T ) N ( U )? Is it possible that they are not equal? Justify your answers. Problem 2. Prove Theorem 2.10 (page 87). Problem 3. This question is regarding the proof of the Dimension Theorem pre- sented during lecture. At what point in the proof was the fact that
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Unformatted text preview: W (the target space) is nite-dimensional used? Why can this assumption be removed (how can the proof be modied to not need it)? Problem 4. Give an example of a linear transformation T : V V such that T is injective (one-one) but not surjective (onto). Hint: V must be innite-dimensional. 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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