UNIVERSITY OF TORONTO MISSISSAUGA AUGUST 2019 FINAL EXAMINATION MAT224H5Y
Linear Algebra II
Ali Mousavidehshikh
Duration – 3 hours
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Please note, once this exam has begun, you CANNOT re-write it. Instructions:
• This exam consists of 7 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 R3 (under the usual operations in R3)?
(1) U = {(x, y, 1) : x = −y}.
(2) U = {(x, y, z) : x + y + z = 1}. (3) U = {(x, 0, z) : x = z}.
(4) None of the above.
(B) Which one of the following sets is linearly independent?
(1) {(1,0,−1,0),(0,1,0,0),(1,0,1,0),(0,0,0,5)} in R4. (2) {x+x2,2x+x3,2×2 −x3} in P3.
(3) {(0, 0, 0), (1, 1, 0).(1, 1, 1)} in R3.
(4) {(1,1,0),(1,1,1),(2,4,−1),(3,2,2)} in R3.
Continued on page 3 . . .
MAT224H5Y EXAM
Page 3 of 17
(C) Suppose U = span{x + x2, 1 + 2x + 2×2, 4, x3 + 4} (a subspace of P3). Then dim U =
(1) 1. (2) 2. (3) 3. (4) 4.
(D) Suppose U and W are subspaces of a vector space V. If dimV = 3, dimU = dimW = 2, and U ̸= W , then dim(U ∩ W ) =
(1) 0. (2) 1. (3) 2. (4) 3.
Continued on page 4 . . .
MAT224H5Y EXAM
Page 4 of 17
(E) IfU={A∈M33 :AT =A},thendimU=
(1) 3. (2) 6. (3) 2. (4) 9.
(F) Suppose T : V → W is an isomorphism. Which one of the following is TRUE?
(1) T(v)=0W ifandonlyifv=0V,where0V and0W arethezerovectorsinV andW,
respectively.
(2) If V = span{v1,v2,…,vn}, then W = span{T(v1),T(v2),…,T(vn)}.
(3) dimV =5ifandonlyifdimW =5.
(4) All of the above.
Continued on page 5 . . .
MAT224H5Y EXAM
(G) If T : P3 → R is given by T (a + bx + cx2 + dx3) = d, then ker T =
(1) span{1, x, x2 }.
(2) span{x, x2 }.
(3) span{1,x}.
(4) {0}, where 0 is the zero polynomial in P3.
Page 5 of 17
(H) Define T : Pn → Pn via T(p(x)) = p(x)+xp′(x) for all p(x) ∈ Pn, where p′(x) is the derivative of p(x). Which one of the following is FALSE?
(1) T is a linear transformation.
(2) ker T = {0}, where 0 is the zero polynomial in Pn. (3) T is onto.
(4) T is not an isomorphism.
Continued on page 6 . . .
MAT224H5Y EXAM
(I) Let V = R3 (with the inner product being the usual dot product), and let
U = span{(1, −1, 0), (−1, 0, 1)}, and v = (2, 1, 0). Then projU (v) =
(1) (1,0,−1).
(2) (0,−1,1).
(3) (2,1,0).
(4) None of the above.
Page 6 of 17
(J) Let P be an orthogonal matrix. Which one of the following is FALSE?
(1) det P = ±1.
(2) IfdetP =−1,thenI+P hasnoinverse. (3) P T P = I , where I is the identity matrix. (4) None of the above.
Continued on page 7 . . .
MAT224H5Y EXAM
Page 7 of 17
(K) Which one of the following is not necessarily true?
(1) Every hermitian matrix is normal.
(2) Every normal matrix is hermitian.
(3) Every unitary matrix is normal.
(4) All the eigenvalues of a hermitian matrix are real.
(L) IfT :P2 →P2 isgivenbyT(p(x))=p(x+1),thendetT =
(1) 1. (2) 3. (3) 4. (4) 2.
Continued on page 8 . . .
MAT224H5Y EXAM
(M) Which one of the following matrices is hermitian?
2−i 6
(3) . 0 1 1+i
0−1+i 3
1 −i1−i
Page 8 of 17
(4) .
i 1 2
(N) Which one of the following is not necessarily true?
(1) If S and T are linear operators on V , where V is finite dimensional, then tr(ST ) = tr(T S).
(2) If S and T are linear operators on V , where V is finite dimensional, then tr(S + T ) = tr(S) + tr(T ).
(3) If V is finite dimensional, the linear operator T on V has an inverse if and only if det T ̸= 0.
(4) If V is finite dimensional, the linear operator T on V has an inverse if and only if tr(T ) ̸= 0.
Continued on page 9 . . .
MAT224H5Y EXAM Page 9 of 17 (O) If B0 denotes the standard basis of R3 and B = {(1, 1, 0), (1, 0, 1), (0, 1, 0)}, then PB0 ←B =
(1) 1 0 1.
1 0 −1.
0 1 0
0 1 0 1 1 0 −1 1 0
(3)1 0 1. (4)1 0 1.
0 1 1 0 −1 0
(P) Which one of the following is an inner product? (1) V =R2,and⟨(a,b),(c,d)⟩=abcd.
(2) V = P3, and ⟨p(x), q(x)⟩ = p(1)q(1).
(3) ⟨v,w⟩ = vTAw for all column vectors v,w ∈ R2, where A = .
(4) V = M22, and ⟨A, B⟩ = det(AB).
5 −3 −3 2
Continued on page 10 . . .
MAT224H5Y EXAM
Page 10 of 17
1 1 1 0 1 0
, , 0 0 1 0 1 1
−1 −1
(1) span . −1 0
0 1
(2)span −1 1
1 − 1 (3) span .
−1 0
1 − 1
(4)span 1 1
in M22 with inner product ⟨A,B⟩ = tr(AB).
(Q) Let U = span Then U⊥ =
(R) Let T be a linear operator on V , and let U and W be T -invariant subspaces of V . Which one of the following is not necessarily true?
(1) U + W is a T-invariant subspace of V . (2) U ∩ W is a T-invariant subspace of V . (3) U ∪ W is a T-invariant subspace of V . (4) T(U)isaT-invariantsubspaceofV.
Continued on page 11 . . .
MAT224H5Y EXAM Page 11 of 17 (S) The norm of f (x) = x in P2 with inner product ⟨p(x), q(x)⟩ = p(0)q(0) + p(1)q(1) + p(2)q(2) is
(1) (2) (3) (4)
(T) Let V = R2 with inner product being usual dot product, and let u = v = (1, 1) and w = (−1, 0). Which one of the following is TRUE?
(1) {u, v, w} is an orthogonal set.
(2) ∥u+v+w∥2 =∥u∥2 +∥v∥2 +∥w∥2. (3) ∥u + w∥2 = ∥u∥2 + ∥w∥2.
(4) All of the above.
Continued on page 12 . . .
MAT224H5Y EXAM Page 12 of 17 Question2.(10Marks)LetU={A∈M22: trA=0}.ProvethatUisasubspaceofM22.
Continued on page 13 . . .
MAT224H5Y EXAM Page 13 of 17 Question 3. Let T : V → W be a linear transformation.
(a) (5 Marks) If T is one to one and {v1,v2,…,vn} is independent in V, prove that {T(v1),T(v2),…,T(vn)} is independent in W.
(b)(5Marks)IfT isontoandV =span{v1,v2,…,vn},provethatW =span{T(v1),T(v2),…,T(vn)}.
Continued on page 14 . . .
MAT224H5Y EXAM Page 14 of 17 Question 4. Define T : Mnn → Mnn via T (A) = A + AT .
(a) (5 Marks) Prove that T is a linear transformation.
(b) (5 Marks) Show that T is not an isomorphism.
Continued on page 15 . . .
MAT224H5Y EXAM Page 15 of 17 Question 5. Let v and w be vectors in an inner product space V .
(a) (5 Marks) Show that v is orthogonal to w if and only if ∥v + w∥ = ∥v − w∥.
(b) (5 Marks) Show that v + w and v − w are orthogonal if and only if ∥v∥ = ∥w∥.
Continued on page 16 . . .
MAT224H5Y EXAM Page 16 of 17
Question 6. (10 Marks) Let V be an inner product space with inner product ⟨ , ⟩, and let T : V → V be an isomorphism. Show that
is an inner product on V .
⟨v,w⟩1 =⟨T(v),T(w)⟩
Continued on page 17 . . .
MAT224H5Y EXAM Page 17 of 17 Question 7. Let B = {f1, f2, . . . , f2} be an orthonormal basis of an inner product space V . Given a
linearoperatorT :V →V,defineT′ :V →V by
T ′(v) = ⟨v, T (fi)⟩fi.
(a) (5 Marks) Prove that (aT )′ = aT ′ for any scalar a.
(b) (5 Marks) Show that MB(T′) is the transpose of MB(T).
End Of Exam
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