Solution Key to Some Exercises in Chapter 3#5. Determine the kernel and range of each of the following linear transformations on (a) (b) (c) Solution (a) Let . . if and only if if and only if . Thus, The range of is (b) Let . . if and only if if and only if and . Thus, The range of is (c) Let . . if and only if if and only if and . Thus, The range of is 备注: 映射的核以及映射的像都是集合,应该以集合的记号来表达或者用文字来叙述. #7. Let be the linear mapping that maps into defined by Find a matrix A such that .Solution Hence, #10. Let be the transformation on defined by a) Find the matrix A representing with respect to b) Find the matrix B representing with respect to c) Find the matrix S such that d) If , calculate .Solution (a) (b) (c) The transition matrix from to is , (d) #11. Let A and B be matrices. Show that if A is similar to B then there exist matrices S and T, with S nonsingular, such that and .Proof There exists a nonsingular matrix P such that . Let , . Then and .#12. Let be a linear transformation on the vector space V of dimension n. If there exist a vector v such that and , show that(a) are linearly independent.(b) there exists a basis E for V such that the matrix representing with respect to the basis E is Proof(a) Suppose that Then That is, Thus, must be zero since . This will imply that must be zero since .By repeating the process above, we obtain that must be all zero. This proves that are linearly independent.(b) Since are n linearly independent, they form a basis for V.Denote . #13. If A is a nonzero square matrix and for some positive integer k, show that A can not be similar to a diagonal matrix.Proof Suppose that A is similar to a diagonal matrix . Then for each , there exists a nonzero vector such that since .This will imply that for . Thus, matrix A is similar to the zero matrix. Therefore, since a matrix that is similar to the zero matrix must be the zero matrix, which contradicts the assumption.This contradiction shows that A can not be similar to a diagonal matrix. Or If then . implies that for . Hence, . This will imply that . Contradiction!4