MA353_HW5_sol

MA353_HW5_sol - Purdue University MA 353: Linear Algebra II...

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Unformatted text preview: Purdue University MA 353: Linear Algebra II with Applications Homework 5, due Feb. 19, Solutions Sec. 2.1 #9 Prove that there exists a linear transformation T : R 2 → R 3 such that T (1 , 1) = (1 , , 2) and T (2 , 3) = (1 ,- 1 , 4). What is T (8 , 11). Proof. Note that the set { (1 , 1) , (2 , 3) } is linearly independent and since dim R 2 = 2, it is also a basis of R 2 . Thus by Theorem 2.6, there is such linear transformation T . Now consider (8 , 11) = 2(1 , 1) + 3(2 , 3) , and so T (8 , 11) = T (2(1 , 1) + 3(2 , 3)) = 2 T (1 , 1) + 3 T (2 , 3) = 2(1 , , 2) + 3(1 ,- 1 , 4) = (5 ,- 3 , 16) . #14: Let T : V → W be linear. (a) Prove that T is one-to-one if and only if T carries linearly indepen- dent subsets of V into linearly independent subsets of W . (b) Suppose that T is one-to-one and that S is a subset of V . Prove that S is linearly independent if and only if T ( S ) is linearly independent. (c) Suppose β = { v 1 ,v 2 ,...,v n } is a basis for V and T is one-to-one and onto. Prove that T ( β ) = { T ( v 1 ) ,T ( v 2 ) ,...,T ( v n ) } is a basis for W . Proof. (a) (= ⇒ ) Assume T is one-to-one. Let S = { v 1 ,v 2 ,...,v n } ⊆ V be linearly independent. We must show that T ( S ) = { T ( v 1 ) ,T ( v 2 ) ,...,T ( v n ) } is linearly independent. So assume a 1 T ( v 1 ) + a 2 T ( v 2 ) + ··· + a n T ( v n ) = 0 ....
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This note was uploaded on 04/29/2010 for the course MA 353 taught by Professor Staff during the Spring '08 term at Purdue.

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MA353_HW5_sol - Purdue University MA 353: Linear Algebra II...

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