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mat67-2011-hw5

mat67-2011-hw5 - by using the following steps(a Denote H =...

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Homework Assignment #5 Math 67 UC Davis, Fall 2011 Homework due. Tuesday 11/01/11 at discussion section. Reading material. Read Sections 6.5–6.6 in the textbook. Problems 1. Compute the coordinate vector [ v ] B , where: (a) v = (1 , 0 , 1), B = { (1 , 0 , 0) , (1 , 1 , 0) , (1 , 1 , 1) } . (b) v = (1 , 0 , 1), B = { (1 , 0 , 1) , (1 , 0 , 0) , (0 , 1 , 0) } . (c) v = z 3 - 2 z , B = { z + 1 , z - 1 , z 2 , z 3 } in the space P 3 of polynomials of degree 3. 2. Compute the representation matrix M ( T ) B C , where: (a) T ( x, y ) = ( x + 10 y, - x ), B = C = { (1 , 0) , (0 , 1) } . (b) T ( x, y, z ) = ( z, y, 3 x ), B = { (1 , 0 , 0) , (0 , 1 , 0) , (0 , 0 , 1) } , C = { (1 , 0 , 1) , (0 , 1 , 0) , ( - 1 , 0 , 1) } (c) T ( x, y, z ) = ( z, y, 3 x ), B = { (1 , 0 , 1) , (0 , 1 , 0) , ( - 1 , 0 , 1) } , C = { (1 , 0 , 0) , (0 , 1 , 0) , (0 , 0 , 1) } 3. Solve calculational exercises 1(d), 1(e), 2(c), 3, 5 in Chapter 6. 4. Let U, V, W be finite-dimensional vector spaces, and let S : U V , T : V W be linear transformations. The goal of this problem is to prove the inequality: dim(null( T S )) dim(null( S )) + dim(null( T )) , where T S : U W denotes the composition of the two transformations. Prove this
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Unformatted text preview: by using the following steps: (a) Denote H = null( T ◦ S ) (a linear subspace of U ), and deﬁne a linear transfor-mation R : H → V by R ( v ) = S ( v ) (i.e., it is the same as S , but its domain is a subspace of the domain of S ; sometimes R deﬁned in this way will be referred to as the restriction of S to H ). Show that null( S ) ⊆ H , and explain why this implies that null( R ) = null( S ). (b) Show that range( R ) ⊆ null( T ). (c) Apply the dimension formula (Theorem 6.5.1 in the textbook) for a suitable linear transformation to deduce the inequality stated at the beginning of the question....
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