UNIVERSITY OF TORONTO MISSISSAUGA AUGUST 2018 FINAL EXAMINATION MAT224H5Y
Linear Algebra II
Ali Mousavidehshikh
Duration – 3 hours
Copyright By PowCoder代写 加微信 powcoder
First name: Student number:
Aids: none
The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, SMART devices, tablets, laptops, calculators, and MP3 players. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over.
If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course.
Please note, once this exam has begun, you CANNOT re-write it. Instructions:
• This exam consists of 6 questions for a total of 100 marks. However, the exam is marked out of 90.
• The exam consists of 17 pages including the cover page. Unless otherwise stated, show your work.
• If you need more space, use the back of a page; clearly indicate when you have done this.
MAT224H5Y EXAM Page 2 of 17
Question 1. (40 Marks) This question consists of 20 multiple choice questions. Answer each question and put your answer in the table below. There is only one correct answer for each question.
(A) Which one of the following is a subspace of P3?
(1) U ={g(x)∈P3 :degg(x)=3}.
(2) U = {xg(x) : g(x) ∈ P3}.
(3) U ={g(x)∈P3 :g(2)=0}.
(4) U={xg(x)+(1−x)h(x):g(x),h(x)∈P3}.
(B) span{(1,0,0),(1,1,0),(1,1,1)}=
(1) span{(0,−1,0),(2,1,1),(1,1,1)}
(2) span{(1,0,0),(1,1,0)}
(3) span{(1,0,0),(1,1,0),(3,2,0),(−1,0,0)} (4) span{(1,1,0),(1,1,1)}
Continued on page 3 . . .
MAT224H5Y EXAM
(C)LetU={A∈Mnn:AT =A}.ThendimU=
(1) n(n+1). 2
Page 3 of 17
(D) Suppose U and W are subspaces of a vector space V such that dimV = 5, dimU = dimW = 3. Which one of the following is impossible?
(1) dim(U∩W)=1 (2) dim(U∩W)=0 (3) dim(U∩W)=3 (4) dim(U∩W)=2
Continued on page 4 . . .
MAT224H5Y EXAM
(E) Which one of the following is a linear transformation?
(1) T : Pn → Pn given by T(p(x)) = p(0).
(2) T :Mnn →RgivenbyT(A)=detA.
(3) T :Mnn →RgivenbyT(A)=rankA.
(4) T :V →V givenbyT(v)=v+u,whereuisafixednon-zerovectorinV.
Page 4 of 17
(F) Suppose T : V → V is a linear transformation such that T(v1) = 1 = T(v2). Then T(v1 +3v2)=
(1) 1 (2) 3 (3) 4 (4) 2
Continued on page 5 . . .
MAT224H5Y EXAM
(G) ThekernelofthelineartransformationT :P2 →R2 givenbyT(a+bx+cx2)=(a,b)is
(2) span{x, x2 }.
(3) span{1,x}.
(4) span{x2}.
Page 5 of 17
(H) Suppose T : V → W is a linear transformation such that kerT = V, where V and W are not the zero vector space. Which one of the following is true.
(1) T is an isomorphism.
(2) T is a bijection.
(3) T is the zero linear transformation. That is, T (x) = 0 for all x ∈ V .
(4) T is the identity linear transformation. That is, T (x) = x for all x ∈ V .
Continued on page 6 . . .
MAT224H5Y EXAM Page 6 of 17
(I) Let T : V → W be a linear transformation, where V and W are finite dimensional vector spaces. Which one of the following is false?
(1) If T is onto, then dimV ≥ dimW.
(2) If T is injective, then dimV ≤ dimW. (3) T is injective if and only if ker T = {0}. (4) If dimV ≥ dimW, then T is onto.
(J) Which one of the following is an isomorphism?
(1) T : R3 → R3 given by T(x,y,z) = (x,y,x).
(2) T : R3 → R3 given by T (x, y, z) = (x, x + y, x + y + z). (3) T : R2 → R2 given by T (x, y) = (x + y, 0).
(4) T : Mnn → Mnn given by T (A) = A − AT .
Continued on page 7 . . .
MAT224H5Y EXAM
Page 7 of 17
(K) If U = span{(1,−1,2,1),(1,0,−1,1),(1,1,1,1)} (a subspace of R4), then dimU =
(1) 1 (2) 3 (3) 2 (4) 4
(L) Suppose A is a square matrix with AAT = I. Which one of the following MUST be true?
(1) 1 is the only eigenvalue of A. (2) A is an orthogonal matrix.
(3) A is symmetric.
(4) −1 is the only eigenvalue of A.
Continued on page 8 . . .
MAT224H5Y EXAM
(M) Which one of the following matrices is Hermitian?
3 2+i (1) .
2−i 6
Page 8 of 17
(3) . 0 i 1
0 −1 3
1 −i1+i
(4) .
i 1 2
(N) If U is unitary, then
(1) UH =U.
(2) UT =U.
(3) UH is unitary. (4) U−1 =U.
Continued on page 9 . . .
MAT224H5Y EXAM
(O) Let V = R3, v = (a,b,c), and B = {(1,−1,2),(1,1,−1),(0,0,1)}. Then CB(v) =
a a−b
1 (1) b. (2) 2 a+b .
c −a+3b+c
a−b a−b
1 (3) a + b . (4) 2 a + b .
−a+3b+2c −a+3b+2c
Page 9 of 17
(P) Define the following inner product on M22: ⟨A, B⟩ = tr(ABT ), where tr stands for trace of a
1 1 1 0 1 0
matrix. If U = span , , , then U⊥ =
0 0 1 0 1 1
1 1
(1)span . 1 0
1 − 1
(2)span . 1 0
− 1
1 .
−1 0
Continued on page 10 . . .
MAT224H5Y EXAM Page 10 of 17
(Q) Let T : P2 → R3 be defined by T(p(x)) = (p(0),p(1),p(2)). Let B = {1,x,x2} and D = {(1,0,0),(0,1,0),(0,0,1)}. Then MDB(T) =
(1) 1 1 1.
1 1 1.
1 2 2 100 100
(3) 1 1 0. (4)0 1 1.
1 0 4 0 2 4
(R) Which one of the following is not a linear transformation?
(1) T : R2 → R2 given by T(a,b) = (−b,−a). (2) T : R2 → R3 given by T(a,b) = (a,−b,0). (3) T : R2 → R2 given by T (a, b) = (0, 0).
(4) T : R3 → R3 given by T(a,b,c) = (a2,b2,c2).
Continued on page 11 . . .
MAT224H5Y EXAM Page 11 of 17 (S) Let B0 = {(1,0,0),(0,1,0),(0,0,1)} and B = {(1,1,0),(1,0,1),(0,1,0)} (bases of R3). Then
(1) 0 1 1.
1 0 1.
(3) 1 1 0.
1 1 1
0 1 1.
0 1 1
(T) Define the following inner product on P2:
⟨p(x), q(x)⟩ = p(0)q(0) + p(1)q(1) + p(2)q(2).
Thenormoff(x)=1+x+x2 is √
Continued on page 12 . . .
MAT224H5Y EXAM Page 12 of 17
Question 2. (10 Marks) Let V = R2. Define addition via (x,y) + (w,z) = (x + w,y + z + 1) and scalar multiplication via a(x, y) = (ax, ay + a − 1). Prove that V is a vector space under these operations.
Continued on page 13 . . .
MAT224H5Y EXAM Page 13 of 17
Question 3. (10 Marks) Let U and W be subspaces of a vector space V , let dim V = n, and assume thatdimU+dimW =n. IfU∩W ={0},provethatV =U⊕W.
Continued on page 14 . . .
MAT224H5Y EXAM
Page 14 of 17
n Question 4. (10 Marks) Suppose {v1,…,vn} is a basis of V. Given u,w ∈ V, u = aivi for
someai ∈R,andw=bivi forsomebi ∈R. Define
Prove that this is an inner product on V .
n ⟨u,w⟩ = aibi.
Continued on page 15 . . .
MAT224H5Y EXAM Page 15 of 17
Question 5. (10 Marks) Let ⟨p(x), q(x)⟩ = p(0)q(0) + p(1)q(1) + p(2)q(2) in P2, and let U = span{1, 1 + x2 } and f (x) = x.
Find the polynomial in U closest to f(x).
Continued on page 16 . . .
MAT224H5Y EXAM Page 16 of 17 Question 6. (20 Marks) Let T : V → V be a linear operator such that T2 = 1V , where dimV = n.
DefineU1 ={u∈V :T(u)=u}andU2 ={u∈V :T(u)=−u}. (a) (5 Marks) Prove that U1 and U2 are subspaces of V .
(b) (5 Marks) Prove that V = U1 ⊕ U2.
Continued on page 17 . . .
MAT224H5Y EXAM Page 17 of 17
Question 6 continued, with notation as listed on the previous page.
(c) (5 Marks) Prove that U1 and U2 are T -invariant.
(d) (5 Marks) Find an ordered basis B of V such that MB(T) = Ik 0 for some integer k, 0 −I
where Ik and In−k are the k × k and (n − k) × (n − k) identity matrices, respectively.
End Of Exam
程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com