Final Exam for Algebra, (Advanced Program K63), 90’
TEST No. 1 (Materials or textbooks are forbidden)
Prob. I. (2.5 Pts). Let n ∈ N and n > 1. Denote by M := {z ∈ C | (z + 2)n = 2(z − 2)n }.
1. Determine M in case n = 5.
z+2 √
∈ n 2}
z−2




1
3a
−4b
−1
−2
 and B :=  6 − 9a .
−1
2
8b
Prob. II. (2.5 Pts). Given a, b ∈ R we consider A :=  2
−2 12a + 3 −16b − 6 −24b − 4
27a − 26
2. Prove that M = {z ∈ C | z 6= 2 and
1. For a = 0 and b = 1 find X ∈ M4×1 (R) such that AX = B.
2. Find the values of a and b such that B ∈
/ Colsp(A).
Prob. III. (2.5 Pts). Let the linear mapping f : R3 → R3 be given by
f (x1 , x2 , x3 ) = (x1 + 2x2 − 3x3 ; −x1 + x2 + 2x3 ; −x1 + 7x2 )
1. Find a basis of the space Imf .
 
1
2. Let A be the matrix representation of f relative to the usual basis of R3 and put B := 2.
3
Prove that the system AX = B has no solution. Then, find the least square solutions of that system.
Prob. IV. (2.5 Pts). Let q : R3 → R be a quadratic form given by q(x, y, z) = 2x2 + 2y 2 + 2z 2 + 6xy + 6yz + 6zx.
1. Transform q to the canonical (diagonal) form.
2. Let A be the matrix representation of q in the usual basis of R3 and B := A2020 + 2018A2019 + A2015 + 2017I.
Find an invertible matrix P such that P −1 BP is a diagonal matrix.
Final Exam for Algebra, (Advanced Program K63), 90’
TEST No. 2 (Materials or textbooks are forbidden)
Prob. I. (2.5 Pts). Let n ∈ N and n > 1. Denote by M := {z ∈ C | (z + 3)n = 3(z − 3)n }.
1. Determine M in case n = 5.
z+3 √
∈ n 3}
z−3




1
4a
−3b
−1
−2
 and B :=  6 − 9a .
−2
2
6b
Prob. II. (2.5 Pts). Given a, b ∈ R we consider A :=  2
−2 16a + 6 −12b − 6 −18b − 4
27a − 26
2. Prove that M = {z ∈ C | z 6= 3 and
1. For a = 0 and b = 1 find X ∈ M4×1 (R) such that AX = B.
2. Find the values of a and b such that B ∈
/ Colsp(A).
Prob. III. (2.5 Pts). Let the linear mapping f : R3 → R3 be given by
f (x2 , x2 , x3 ) = (x1 − 2x2 + 3x3 ; x1 − x2 + 2x3 ; x1 + x3 )
1. Find a basis of the space Imf .
 
1
2. Let A be the matrix representation of f relative to the usual basis of R3 and put B := 2.
4
Prove that the system AX = B has no solution. Then, find the least square solutions of that system.
Prob. IV. (2.5 Pts). Let q : R3 → R be a quadratic form given by q(x, y, z) = 3x2 + 3y 2 + 3z 2 + 8xy + 8yz + 8zx.
1. Transform q to the canonical (diagonal) form.
2. Let A be the matrix representation of q in the usual basis of R3 and B := A2020 + 2019A2018 + A2016 + 2015I.
Find an invertible matrix P such that P −1 BP is a diagonal matrix.