LINEAR ALGEBRA - MATH 201
FINAL EXAM - 14/01/2023
Full Name
Student Number
Signature
Duration: 120 minutes. There are 6 problems worth a total of 75 points.
Instructor: Canan Kaşıkcı
Rules
• All unauthorized materials (textbooks, notes, electronic devices, bags, etc.), and in particular mobile
phones and smart watches must be switched off and must be placed on the floor.
• ANY form of communication with fellow students during the exam will not be accepted. Students
involved will immediately be expelled from the classroom.
• If you leave the classroom after the exam has started then you are not allowed back in.
• There are 6 questions worth a total of 75 points (70+5 bonus points.) Check whether your exam
sheet has all of them. If any is missing report this to the proctor.
• You are not allowed to ask questions during the last 30 minutes.
Attention
• Read the questions carefully.
• In your solutions, explain the arguments leading to the answer in a clear, organized fashion. You
might get no points for correct answers with no explanation and/or unorganized solution. No work/explanation equals no credit.
• Note that parts (a) and (b) are independent of each other in Problem 3.
• It is straightforward to find the eigenvalues in Problem 5, hint is given to make sure that one
finds and uses the correct eigenvalues in the rest of the problem. One must use the
cofactor expansion formula and state which row or column is used in the expansion.
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Problem 1: (13 points)
Consider the following linear system of equations:
x+y−z =3
2x + y + z = 2
x + 2y − k 2 z = k + 5
Use Gaussian elimination to find the values of k for which the resulting system has:
i) no solutions;
ii) unique solution;
iii) infinitely many solutions. In this case find the solution set of the linear system.
(Warning: One must justify the answers in all cases above. Note that k values are integers.)
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Name, Surname:
Problem 2: (13 points)

−2 −5
8
 1
3
−5
a) The matrix A = 
 3 11 −19
1
7 −13
Signature:

0 1 0 1
1 −2 0 3 
.
0 0 1 −5 
0 0 0 0


1
0 −17


0
1
5 
reduces to the matrix R = 
 0
7
1 
0
5 −3
i) Determine a basis for Row(A). Explain briefly.
ii) Determine a basis for Null(A).
iii) If W = Row(A), then dim(W ⊥ ) =
since
.


3
 2 

b) Let B be a 3 × 4 matrix with columns C1 , C2 , C3 , C4 . Suppose the vector x = 
 1  ∈ Null(B) and
 
 
1
1
4
the vectors v1 =  2  and v2 =  1  span Col(B).
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i) Express C4 as a linear combination of C1 , C2 , C3 .
ii) Determine rank(B) and dim(Null(B)). Justify your answer.
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Problem 3: (14 points)
a b
a) Let T : M2×2 (R) → R2 [x] be the linear map given as T
= (a + b) + (b + c)x + (c + d)x2 .
c d
i) Determine a basis for Ker(T) and find nullity of T.
ii) State the dimension Theorem for T and use it to find the rank of T.
iii) Determine whether T is injective and surjective.
b) Let B = [1, 1 + x, x + x2 , x + x3 ] be an ordered basis of R3 [x]. Let T : R3 [x] → R3 [x] be linear.
i) Find the coordinate vector (v)B of v = 2 + 2x − x2 + x3 relative to the basis B. (Look careful,
compute little!)
ii) If 1 + x is an eigenvector of T corresponding to the eigenvalue 3, x + x2 ∈ Ker(T) and x + x3
is an eigenvector of T corresponding to the eigenvalue 4, compute T (2 + 2x − x2 + x3 ).
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Problem 4: (14 points)
Let T : R2 [x] → R2 [x] be the linear operator defined as
T (a + bx + cx2 ) = (a − b + c) + (2b − 4c)x + (4c)x2 .
a) Find the matrix [T ]B of T relative to the basis B = [1, x, x2 ].
b) Is T injective? Explain briefly.
c) Find the characteristic polynomial PT (λ) of T and determine the eigenvalues of T.
d) Is T diagonalizable? Explain briefly.
e) Find a basis for the eigenspace of [T ]B corresponding to the eigenvalue λ = 4 and use it to find a
basis for ET (4), i.e. the eigenspace of T corresponding to the eigenvalue λ = 4.
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Problem 5: (16 points)


5 2 0
a) Find the characteristic polynomial PA (λ) and determine the eigenvalues of A =  1 6 0  .
0 0 7
(Hint: Eigenvalues are 4 and 7.)
b) Find a basis for EA (4), i.e. the eigenspace of A corresponding to the eigenvalue λ = 4.
c) Find a basis for EA (7), i.e. the eigenspace of A corresponding to the eigenvalue λ = 7.
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Name, Surname:
Signature:
d) Use test for diagonalization to verify that A is diagonalizable.
e) Find a matrix P and a diagonal matrix D such that P D = AP holds.
Problem 6: (5 points)
(Bonus Problem:) Let u be a fixed vector in Rn and let W be a subspace of Rm . Show that
S = {A ∈ Mm×n (R)|Au ∈ W }
is a subspace of Mm×n (R).
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