8.1 WRITE on paper for upvote. Section: 25, 32, 33 please

8.1 General Linear Transformations 20. Consider the basis S = {V1, V2} for R', where v, = (-2, 1) and 9. T : Rx - R., where V2 = (1, 3), and let T : R2 - R' be the linear transformation T(do, dj , a2, . . ., an) . .. ) = (0, do, a1, a2, . . ., an; . . . ) such that 10. Let T : P2 - P. be the linear transformation defined by T(v,) = (-1,2,0) and T(v2) = (0, -3, 5) T(p(x)) = xp(x). Which of the following are in ker(T)? Find a formula for T(x, , X2), and use that formula to find a. x b. 0 C. 1+ x d. -x T(2, -3). 11. Let T : P2 - P, be the linear transformation in Exercise 10. 21. Consider the basis S = {v, , V2, V, }for R3, where v, = (1, 1, 1), Which of the following are in R(T)? V, = (1, 1,0), and v, = (1, 0, 0), and let T' : R3 - R3 be the a. x+ x2 b. 1 +x c. 3- x2 d. -x linear operator for which 12. Let V be any vector space, and let T : V - V be defined by T(v,) = (2, -1,4), T(V2) = (3, 0, 1), T(v) = 3v. T(v;) = (-1,5,1) a. What is the kernel of T? Find a formula for T(x1, X2, X;), and use that formula to find b. What is the range of T? T(2,4, -1). 13. In each part, use the given information to find the nullity of 22. Consider the basis S = {v, , V,, v, } for R*, where v, = (1, 2, 1), the linear transformation T. V, = (2,9,0), and v, = (3, 3, 4), and let T : R - R' be the a. T : R' - P, has rank 3. linear transformation for which b. T : P. - P, has rank 1. T(v, ) = (1,0), T( V2 ) = (-1, 1), T(V3) = (0,1) c. The range of T : MmMM - R' is R3. Find a formula for T(X1 , X2,X3), and use that formula to find T (7, 13, 7). d. T : M2 - Mazz has rank 3. 14. In each part, use the given information to find the rank of the In Exercises 23-24, let T be multiplication by the matrix A. Find linear transformation T a. a basis for the range of T. a. T : R' - M32 has nullity 2. b. a basis for the kernel of T. b. T : P3 - R has nullity 1. c. the rank and nullity of T. C. The null space of T : Ps - P, is Ps. d. the rank and nullity of A. d. T : Pn - Mmm has nullity 3. 15. Let T : M22 - M22 be the dilation operator with factor k = 3. 23. A = 5 6 -4 24 . A = 4 0 - 2 a. Find 7( 3]) In Exercises 25-26, let TA : R - R3 be multiplication by A. Find a b. Find the rank and nullity of T. basis for the kernel of TA, and then find a basis for the range of TA that consists of column vectors of A. 16. Let T : P2 -P2 be the contraction operator with factor k = 1/4. a. Find T(1 + 4x + 8x?). 25. A = = 1 31 b. Find the rank and nullity of T. 17. Let T : P2 - R3 be the evaluation transformation at the 26. A = -2 4 2 2 sequence of points -1, 0, 1. Find a. T(x?) b. ker(T) C. R(T) 27. Let T : P; - P, be the mapping defined by 18. Let V be the subspace of C[0, 27 ] spanned by the vectors 1, T ( do + a, x+ agx? + agx' ) = 500+ a,x2 sin x, and cosx, and let T : V - R' be the evaluation transfor- mation at the sequence of points 0, 7, 27. Find a. Show that T is linear. b. ker(T) b. Find a basis for the kernel of T. a. T(1 + sinx + cosx) c. Find a basis for the range of T. c. R(T) 28. Let T : P2 - P, be the mapping defined by 19. Consider the basis S = {v1, V2} for R?, where v, = (1, 1) and V2 = (1,0), and let T : RZ - RZ be the linear operator for T (do + ax + ax") = 300 + ax+ (do + a, )x? which T(v,) = (1, -2) and T(v2) = (-4, 1) a. Show that T is linear. Find a formula for T(x,, *2), and use that formula to find b. Find a basis for the kernel of T. I ( 5 , - 3). c. Find a basis for the range of T. 458 CHAPTER 8 General Linear Transformations 29. a. (Calculus required) Let D : P; - P2 be the differentia- tion transformation D(p) = p' (x). What is the kernel of D? 37. Let {V1, V2. . ... V.} be a basis for a vector space V, and let T : V - V be a linear operator. Prove that if b. (Calculus required) Let J : P, - R be the integration transformation J (p) = S_, p(x) dx. What is the kernel of ? T(v, ) = VI, T(V2 ) = v2.... T(vn) = Vn 30. (Calculus required) Let V = C[a, b] be the vector space of then T' is the identity transformation on V. continuous functions on [a, b], and let T : V - V be the transformation defined by 38. Prove: If {v1, V2, . . .. Vn) is a basis for a vector space V and W1, W2. . .., W, are vectors in a vector space W, not neces T (f ) = 5f (x) + 3 / f( 1 )at sarily distinct, then there exists a linear transformation T that maps V into W such that Is T a linear operator? 31. (Calculus required) Let V be the vector space of real-valued T( v, ) = WI, T(v2 ) = wz,..., T(vn) = W. functions with continuous derivatives of all orders on the 39. Let qo (x) be a fixed polynomial of degree m, and define a func- interval (-co, co), and let W = F(-co, co) be the vector tion T with domain Pr by the formula T(p(x)) = P(q.(x)). space of real-valued functions defined on (-co, Do). Prove that T is a linear transformation. a. Find a linear transformation T : V -W whose kernel is P3. True-False Exercises b. Find a linear transformation T : V - W whose kernel TF. In parts (a)-(i) determine whether the statement is true or is Pr. false, and justify your answer. 32. For a positive integer n > 1, let T : Man - R be the linear a. If T(CV, + C2V2) = G, T(v,) + CT(v2) for all vectors v, transformation defined by T(A) = tr(A), where A is an n x n and V2 in V and all scalars c, and c2, then T is a linear matrix with real entries. Determine the dimension of ker(T). transformation. 33. a. Let T : V - R3 be a linear transformation from a vector b. If v is a nonzero vector in V, then there is exactly one lin- space V to R'. Geometrically, what are the possibilities for ear transformation T : V - W such that the range of T? T(-V) = -T(v) b. Let T : R3 - W be a linear transformation from R' to a vec- tor space W. Geometrically, what are the possibilities for c. There is exactly one linear transformation T : V - W for the kernel of T? which T(u + v) = T(u - v) for all vectors u and v in V. 34. In each part, determine whether the mapping T : P, - Pr, is d. If vo is a nonzero vector in V, then T(v) = Vo + v defines linear. a linear operator on V. a. T(p(x)) = p(x+1) e. The kernel of a linear transformation is a vector space. b. T(p(x)) = p(x)+1 35. Let v1, V2, and v3 be vectors in a vector space V, and let f. The range of a linear transformation is a vector space. T : V - R' be a linear transformation for which g. If T : P6 - M2 is a linear transformation, then the nul- T(v,) = (1, -1, 2), T(V2) = (0, 3, 2), lity of T is 3. T (v3) = (-3, 1, 2) h. The function T : M22 - R defined by T(A) = det A is a Find T(2V, - 3V2 + 4V3). linear transformation. Working with Proofs i. The linear transformation T : M22 - M22 defined by 36. Let {V1, V2, . .., Vn} be a basis for a vector space V, and let T : V - W be a linear transformation. Prove that if T( A) = 1 3A T (v1 ) = T(V2 ) =... = T(vn) = 0 then T is the zero transformation. has rank 1