Q7 - stc ksa iı.howard_anton_chris_rorres_ele... ¿JI >written as a linear combination of the basis vectors (v.....,), saykei+..+k,v, = k +..+k,v,Thuski +.+k,v, -k, --k =0Since (v....) is linearly independent, all of the ks are zero; in particular,k =.= k, = 0, which completes the proof. 4456 Chapter 8 General Linear TransformationsExercise Set 8.1In Exercises I-2, suppose that T is a mapping whose domainis the vector space Ma. In cach part, determine whether Tisalinear transformation, and if so, find its kernel.(c) The range of T: M R' is R(d) T: M-M has rank 3.14. In cach part, use the given information to find the rank of thelinear transformation T1. (a) TA) A(b) 7(A) tr(A)(a) T:R-- My has nullity 2.(b) T:P-Rhas nullity I.(c) TLA) = A +A2. (a) TA)- (A)u(b) T(A)=0t.(c) TUA) = cA(c) The null space of T:PP, is P(d) T:P.-M.. has mullity 3.In Exercises 9, determine whether the nupping Tis a lincartransformation, and if so, find its kernel.1T:R'-R. where Tu) = |lul.15. Let T:Ma M be the dilation operator with factor k-3(a) Find 74T:R'R', where v, is a fixed vector in R' andTiw) x(b) Find the rank and nullity of T.S.T:Ma- Ma. where i is a fixed 2 x 3 matrix andTA)= AR.16. Let T:P- be the contraction operator with factorA- 1/4,(a) Find 7(1+ 4x + &x).6. T:M-R, where(b) Find the rank and nullity of T.(a) T|3a - 46+e-d17. Let T:P-R' be the evaluation transformation at the se-quence of points -1.0, 1. Find(a) T)(b) T(b) ker(7)(c) R(T7. T:P-P, where(a) Tia, + dx + ar) = d + a(x +1)+atr+1)(b) T, +aa + aa)18. Let V be the subspace of C10, 2r| spanned by the vectors I.sin x, and cos a, and let T:V-R be the evaluation trans-formation at the sequence of points 0, r, 2. Find(a) 7(1+ sin x+ cos)= (a, + 1)+(a + x + (a + Dr(b) ker(T)(c) RIT)8.T:F(-, )-F(-, *), where(a) T)-1+ fu)(b) 7) fx+1)19. Consider the basis S = tv. vl for R". where v, (1. 1) andv: = (1.0). and let T:R-R be the linear operator forwhich9. T:RR.whereTi, . )- (0, e ..le)T) = (1,-2) and 7v) - (-4, 1)Find a formula for T. A), and use that formula to findTIS,-3)10. Let T:P P, be the linear transformation defined byTip) - xp(x). Which of the following are in ker(7)?(e) 1+x(a)II. Let T: P, be the linear transformation in Exetcise 10.Which of the following are in R(TY?20. Consider the basis S= (v. val fon R', where v, (-2. 1)and- (1, 3), and let T:R-R' be the lineur transformation(b) 0(d) -such thatTi) = (-1.2.0) and 7)- (0,-3. 5)(a) x +x (b) 1+* (e 3-(d) -x12. Let V be any vector space, and let T:VV be defined byFind a formula for T. A), and use that formula to findT2,-3).Ti) = 321. Consider the basis S- (. v, v) for R'. wherev-(1, 1. 1), v (1. 1,0). and (1,0,0). and letT:R'R' be the linear operator for which(a) What is the kernel of T(b) What is the range of T?13. In each part. use the given information to find the nullity ofTiv) = (2. -1. 4). T) - 3,0, 1).the linear transformation TTiv) = (-1. 5. 1)(a) T:R- P, has rank 3.(b) T:P,- P, has rank 1.Find a formula for Ti, ), and use that formula to findT2. 4. -1).8.1 General Linear Transformations 45730. In cach part, determine whether the mapping T:P,-P, is22. Consider the basis S= (v. ) for R', where= (1,2, ), (2.9, 0), and (3, 3, 4), and letT:R'R be the linear transformation for whichlinear.(a) T(p(x)) pLx + 1)Ti) = (1.0). Tiv:) =(-1. I). Tiv) = (0, 1)(b) T(pu)) = PX) +1Find a formula for Tn. .), and use that formula to find M. Let , , and v, be vectors in a vectot space V, and letT:V-R' be a linear transformation for which

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Description: Q7 - stc ksa iı.howard_anton_chris_rorres_ele... ¿JI >written as a linear combination of the basis vectors (v.....,), saykei+..+k,v, = k +..+k,v,Thuski +.+k,v, -k, --k =0Since (v....) is linearly independent, all of the ks are zero; in particular,k =

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Algebra

Exercise Set 8.1 In Exercises 1-2, suppose that Tis a mapping whose domain is the vector spuce Ma. In each part, determine whether T isa linear transformation, and if so, find its kernel. L. (a) TA)- A (e) TLA)A+A (b) 7(A) - A) 2 (a) TLA) = (A) (e) TA) - CA (b) T(A) - 0 In Exercises, determine whether the mapping T is a linear transformation, and if so, find its kernel. A T:RR, where Tia)-u. 4T:RR', where , is a fised vector in R' and Tim) X S.T: Ma- Ma. where i iwa fisod 2 x I matrix und TA)= AR. 6.T:M-R. where (a) 7 (b) T 7. T:P-P, where (a) Tia, + x +aty)d +a,+1) +atr+1) (b) T, + + ax) (a, +)+ (a + Dx + (a, +

Exercise Set 8.1 In Exercises 1-2, suppose that Tis a mapping whose domain is the vector spuce Ma. In each part, determine whether T isa linear transformation, and if so, find its kernel. L. (a) TA)- A (e) TLA)A+A (b) 7(A) - A) 2 (a) TLA) = (A) (e) TA) - CA (b) T(A) - 0 In Exercises, determine whether the mapping T is a linear transformation, and if so, find its kernel. A T:RR, where Tia)-u. 4T:RR', where , is a fised vector in R' and Tim) X S.T: Ma- Ma. where i iwa fisod 2 x I matrix und TA)= AR. 6.T:M-R. where (a) 7 (b) T 7. T:P-P, where (a) Tia, + x +aty)d +a,+1) +atr+1) (b) T, + + ax) (a, +)+ (a + Dx + (a, +

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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