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Math / Linear Algebra 4 / Chapter 2.4 / Problem 24

Linear Algebra | 4th Edition | ISBN: 9780130084514 | Authors: Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence

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Let T: V Z be a linear transformation of a vector space V onto a vector space Z. Define

Chapter 2, Problem 24

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QUESTION:

Let T: V Z be a linear transformation of a vector space V onto a vector space Z. Define the mapping T: V/N(T) - Z by T(v + N(T)) = T(v) for any coset v + N(T) in V/N(T). (a) Prove that T is well-defined; that is, prove that if v + N(T) = v' + N(T), then T(v) = T(v'). (b) Prove that T is linear. (c) Prove that T is an isomorphism. (d) Prove that the diagram shown in Figure 2.3 commutes; that is, prove that T = Tr/.

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QUESTION:

Let T: V Z be a linear transformation of a vector space V onto a vector space Z. Define the mapping T: V/N(T) - Z by T(v + N(T)) = T(v) for any coset v + N(T) in V/N(T). (a) Prove that T is well-defined; that is, prove that if v + N(T) = v' + N(T), then T(v) = T(v'). (b) Prove that T is linear. (c) Prove that T is an isomorphism. (d) Prove that the diagram shown in Figure 2.3 commutes; that is, prove that T = Tr/.

ANSWER:

Step 1 of 5

Let be a linear transformation from the vector space V onto the vector space Z. The mapping is defined by

To prove that is well defined.

Since T is a linear transformation, T is well defined. Therefore, for ,

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