MATH 219 Final, Fall 2004

AMERICAN UNIVERSITY OF BEIRUT Mathematics Department Math 219 - Final Fall 2005-2006 Name:.................................. ID:................................ Instructor: Ms. Diana Audi Time: 2 hrs I- (20 points) Let V = C[0, 1] be the vector space of all continuous functions on [0, 1]. We define an inner product < ., . > on V to be: < f, g >= � 1 0 f(x)g(x)dx, forallf, g ∈ V. a- Let W = span{1, x}, h = 4 + x2, find the orthogonal projection of h on W. b- Find the distance d(h, W) from h to W. c- Find an upper bound on � 1 0 x100 √ x2 + 1dx. II- (35 points)Let T : M2×2(F) → M2×2(F) be the mapping defined by T(A) = AT, the transpose of A, and F is the field of complex numbers. a- Verify that T is a linear operator on M2×2(F). b- Let β = {e1, e2, e3, e4} be the standard basis for M2×2(F) ,find [T]β . c- Find the eigen values of T and describe the corresponding eigen spaces. d- Show that T is digonalizable, and find a basis β′ of M2×2(F) consisting of eigen vectors of T. e- Find an invertible matrix P such that [T]β′ = P −1[T]βP. III- (25 points)Let An×n matrix ,with aij = 1, for all i, j. A =           1 1...... 1 1 1...... 1 1 1...... 1 . ....... . . ....... . 1 1....... 1           a- Find rank(A) and nullity(A). b- Deduce that λ1 = 0 is an eigen value of A,and find the dimension of eigen space corresponding for λ1 = 0. c- Show that λ2 = n is an eigen value of A,and find the dimension of eigen space corresponding for λ2 = n.(Hint: find A           1 1 1 . . 1           ) 1 d- Show that A is diagonalizble . IV- (25 points) Let T : F 3 → F 3 be a linear operator on F 3, where F is the field of complex numbers. Let β = {e1, e2, e3} be the standard basis of F 3. Let [T]β =    2 0 i 0 1 0 −i 0 −2    a- Show that T is a self adjoint linear operator. b- Deduce that T is diagonalizable. c- Find the eigen values of T. d- let ||T|| = max||x=1|| ||T(x)||, Find ||T||? V- (25 points)Let V be an inner product space over R .Let T : V → V be a linear operator on V. a- Show that < x, y >= 1 4(||x + y||2 − ||x − y||2). b- Using part(a), show that ||T(x)|| = ||x|| for all x ∈ V if and only if < T(x), T(y) >=< x, y > for all x, y ∈ V. c- If T is an isometry i.e (||T(x)|| = ||x|| for all x ∈ V ), find the eigen values of T. d- If T is an isometry ,show that T is one to one. VI- (15 points) 1- Let T : V → W be a linear transformation, with dim(V ) = n Let {v1, ..vk} be a basis for kerT, and {v1, ...vk, vk+1, ..vn} a basis for V. a- Show that {T(vk+1), .., T(vn)} is a basis for the range of T. b- Deduce that Rank(T) + N(T) = n 2-(15 points)- Let T : V → V be a linear operator, with dim(V ) = n and rank(T 2 = ToT) = rank(T). a- show that N(T 2) = N(T). b- Deduce that KerT 2 = KerT. c- Show that V = KerT ⊕ Range(T). Application:( 25 points) Let V be an inner product space, and W be a subspace of V. Let T : V → V be a linear operator on V, with T(v) = projW(v) for all v ∈ V. a- Find range(T) and Ker(T). d- Find T 2 = ToT. c- Show that ||T(v)|| ≤ ||v|| for all v ∈ V. VII- (15 points)Let T : R3 → R3 be a linear operator on R3, with T 3 = ToToT = 0. Let a ∈ R3 be such that T 2(a) ̸= 0. Show that {a, T(a), T 2(a)} is a basis for R3. 2 VIII- (Bonus 10 points)Let T, S be linear operators on a finite dimensional vector space V. a- If ToS is one to one, prove that S is one to one. b- If ToS is onto ,prove that T is onto. c- Deduce that if ToS is invertible ,then T and S are invertible. 3