4. Let T: VW be a linear transformation between finite-dimensional inner productspaces, and let T*: W→ V be the adjoint of T. Prove the following statements.1=(a) rank(T*T) = rank(T). (Hint: N(T*T) =Sec. 6.3 Exercise 15(e).)(b) T is injective if and only if T*T is invertible. (Hint: use (a).)(c) rank(TT*) rank(T). (Hint: interchange the roles of T and T* in (a), andrecall that rank(T*) = rank(T).)=N(T) by Homework 9 Textbook

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Answered: 4. Let T: V→ W be a linear…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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ST])= (a - b, cd). Determine whether or not T' is a linear 3. Let T: M22 - R be defined as T transformation, then prove your claim.
+ Crun) 2. You are given that T : U→ X is a linear transformation. Prove that, T(c,u, + c2U2 + = c, T(u,) + C2T(u2) + c,T(un) for all u; e U and for all c; e R and n2 1. +..
3 be W be vector a positive integer, V and field F, T: VV be linear transformation, and Spaces over f: V²->w be a multilinear alternating function. Let g: V" w be the function defined by Let n a V g (V., ..., Vn ) = + (T(V₁), T(V₂), ...,T (Un)) (V.... . Vn) EV" Prove the following statements. (a g is multilinear and alternating. (b) If V=F", W=F₁, and T=LA for some A € Mnxn (F), then g= (detA/.f
+b] Let T : R? → R be a linear transformation. Which of the following is equal to T a +b O A. T (B) +" (!) В. T +T +T +T Oc. T 피 (B) OD. | 2T
Let T : U → V be a linear transformation. Use the rank-nullity theorem to complete the information in the table below. U R R" dim(U) 5 Ex: 5 Ex: n+2 rank(T) nullity(T) 4 Ex: 5 Ex: n+2 Ex: 5 6 5
Let U,V ,W be finite-dimensional F-vector spaces and let T:U → V, S:V→W be linear transformations. Prove that nullity(S • T) - nullity(T ) = dim(im(T ) N ker(S)) following steps (a)-(c) below. a) LetU:=ker(S-T)andT:=T|u.Prove:im(T').=T(U')=im(T)N ker(S ). b) Prove:ker(T')=ker(T). c) Deduce:nullity(S•T)-nullity(T)=dim(im(T)nker(S)).
4. Let VCR" be a subspace. Let F: V→V and G: V → V be invertible linear transformations. Denote by F¹: V→V and G¹: V → V the inverses of F and G respectively. Is the map H: V → V defined by H(v):= F(G(F¹(G¹(v)))), for v € V, a linear transformation? Justify your answer.
Consider two finite dimensional (real) vector spaces V and W with dim(V) = n and dim(W) = k. Define L(V, W) to be the set of linear transformations: L(V, W) = {T: The map T: VW is linear} 1. Define addition and scaling on L(V, W). 2. Sketch a proof that L(V, W) is a vector space. 3. Prove that L(V, W) is finite dimensional by finding an explicit basis for it.
Let f : R? → R be defined by f((x, y)) = 5x + 7y. Is f a linear transformation? a. f({x1,Yı) + (x2, Y2)) : f({x1, Y1)) + f({x2, Y2)) = + Does f((x1,Y1) + (x2, Y2)) = f(x1, Y1)) + f((x2, Y2)) for all (01, Y1), (x2, Y2) E R?
Q3. Let A = -1 1 2 0 3 -2 0-6 4 0 12 (a) Find the null space of A. (b) Let a linear transformation T: R → R be defined by T() = Au. Find n and m. (c) For the linear transformation T defined in part (b) find dim(Ker(T)) and dim(Im(T)).
Let T:R^3⟶R^3 be a linear operator defined by T([(x,y,z)])= [(x-y,-z,2z)] a) Show that T is a linear transformation. b) Describe R(T). What is the dimension of R(T)? c) Find a basis for the null space of T.
a) let T : R3 → R be a linear transformation defined by T(a, b, c) = 2a−b+ 5c. Find y ∈ R3such that T(x) =< x, y > for all x ∈ R3.b) Let T : V → V be a linear operator over a finite dimensional vector space V . Prove thatdim(ker(T)) + dim(ImmT) = Dim(V )

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