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Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms.
In Exercises 33 through 36, V denotes the space of infinite sequences of real numbers.
36.
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Linear Algebra with Applications (2-Download)
- Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].arrow_forward27. Let , where and are nonempty. Prove that has the property that for every subset of if and only if is one-to-one. (Compare with Exercise 15 b.). 15. b. For the mapping , show that if , then .arrow_forwardFind the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).arrow_forward
- 23. Let be the equivalence relation on defined by if and only if there exists an element in such that .If , find , the equivalence class containing.arrow_forwardA relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.arrow_forwardLet T be a linear transformation from P2 into P2 such that T(1)=x,T(x)=1+xandT(x2)=1+x+x2. Find T(26x+x2).arrow_forward
- (See exercise 24.) Show that the relation f(x)Rg(x) if and only if f(x)=g(x) is an equivalence relation on R[ x ]. Describe the equivalence class [ f(x) ]. For each f(x)=i=0naixi in R[ x ], the formal derivative of f(x) is the polynomial f(x)=i=1niaixi1. (For n=0, f(x)=0 by definition.) Prove that [ f(x)+g(x) ]=f(x)+g(x). Prove that [ f(x)g(x) ]=f(x)g(x)+f(x)g(x).arrow_forwardDescribe the kernel of epimorphism in Exercise 20. Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].arrow_forward5. For each of the following mappings, determine whether the mapping is onto and whether it is one-to-one. Justify all negative answers. (Compare these results with the corresponding parts of Exercise 4.) a. b. c. d. e. f.arrow_forward
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