Seminar 4

1. Let K be a field. Show that K[X] is a K-vector space, where the addition is the
usual addition of polynomials and the scalar multiplication is defined as follows: ∀ k ∈ K,
∀f = a0 + a1 X + · · · + an X n ∈ K[X],

k · f = (ka0 ) + (ka1 )X + · · · + (kan )X n .

2. Let K be a field and m, n ∈ N, m, n ≥ 2. Show that Mm,n (K) is a K-vector space,

with the usual addition and scalar multiplication of matrices.
3. Let K be a field, A ̸= ∅ and denote K A = {f | f : A → K}. Show that K A is
a K-vector space, where the addition and the scalar multiplication are defined as follows:
∀f, g ∈ K A , ∀k ∈ K, f + g ∈ K A , kf ∈ K A ,

(f + g)(x) = f (x) + g(x) , (k · f )(x) = k · f (x) , ∀x ∈ A .

4. Let V = {x ∈ R | x > 0} and define the operations: x ⊥ y = xy and k ⊺ x = xk ,

∀k ∈ R and ∀x, y ∈ V . Prove that V is a vector space over R.
5. Let K be a field and let V = K × K. Decide whether V is a K-vector space with
respect to the following addition and scalar multiplication:
(i) (x1 , y1 )+(x2 , y2 ) = (x1 +x2 , y1 +2y2 ) and k ·(x1 , y1 ) = (kx1 , ky1 ), ∀(x1 , y1 ), (x2 , y2 ) ∈
V and ∀k ∈ K.
(ii) (x1 , y1 )+(x2 , y2 ) = (x1 +x2 , y1 +y2 ) and k·(x1 , y1 ) = (kx1 , y1 ), ∀(x1 , y1 ), (x2 , y2 ) ∈ V
and ∀k ∈ K.
6. Let p be a prime number and let V be a vector space over the field Zp .
(i) Prove that x + · · · + x = 0, ∀x ∈ V .
| {z }
p times
(ii) Is there a scalar multiplication endowing (Z, +) with a structure of a vector space
over Zp ?
7. Which ones of the following sets are subspaces of the real vector space R3 :
(i) A = {(x, y, z) ∈ R3 | x = 0};
(ii) B = {(x, y, z) ∈ R3 | x = 0 or z = 0};
(iii) C = {(x, y, z) ∈ R3 | x ∈ Z};
(iv) D = {(x, y, z) ∈ R3 | x + y + z = 0};
(v) E = {(x, y, z) ∈ R3 | x + y + z = 1};
(vi) F = {(x, y, z) ∈ R3 | x = y = z} ?
8. Which ones of the following sets are subspaces:
(i) [−1, 1] of the real vector space R;
(ii) {(x, R2 | x2 + y 2 ≤ 1} of the real vector space R2 ;
y) ∈ 
n
a b
o
(iii) a, b, c ∈ Q of Q M2 (Q) or of R M2 (R);
0 c
(iv) {f : R → R | f continuous} of the real vector space RR ?
9. Which ones of the following sets are subspaces of the K-vector space K[X]:
(i) Kn [X] = {f ∈ K[X] | degree(f ) ≤ n} (n ∈ N);
(ii) Kn′ [X] = {f ∈ K[X] | degree(f ) = n} (n ∈ N).
10. Show that the set of all solutions of a homogeneous system of two equations and
two unknowns with real coefficients is a subspace of the real vector space R2 .

1
Seminar 5
1. Determine the following generated subspaces:
1, X, X2 >
(i) <  in the
 real
 vector
 space
R[X].
1 0 0 1 0 0 0 0
(ii) , , , in the real vector space M2 (R).
0 0 0 0 1 0 0 1
2. Consider the following subspaces of the real vector space R3 :
(i) A = {(x, y, z) ∈ R3 | x = 0};
(ii) B = {(x, y, z) ∈ R3 | x + y + z = 0};
(iii) C = {(x, y, z) ∈ R3 | x = y = z}.
Write A, B, C as generated subspaces with a minimal number of generators.
3. Consider the following vectors in the real vector space R3 :

a = (−2, 1, 3), b = (3, −2, −1), c = (1, −1, 2), d = (−5, 3, 4), e = (−9, 5, 10).

Show that ⟨a, b⟩ = ⟨c, d, e⟩.

4. Let
S = {(x, y, z) ∈ R3 | x + y + z = 0},
T = {(x, y, z) ∈ R3 | x = y = z}.
Prove that S and T are subspaces of the real vector space R3 and R3 = S ⊕ T .
5. Let S and T be the set of all even functions and of all odd functions in RR respectively.
Prove that S and T are subspaces of the real vector space RR and RR = S ⊕ T .
6. Let f, g : R2 → R2 and h : R3 → R3 be defined by

f (x, y) = (x + y, x − y),

g(x, y) = (2x − y, 4x − 2y),

h(x, y, z) = (x − y, y − z, z − x).
Show that f, g ∈ EndR (R2 ) and h ∈ EndR (R3 ).
7. Which ones of the following functions are endomorphisms of the real vector space R2 :
(i) f : R2 → R2 , f (x, y) = (ax + by, cx + dy), where a, b, c, d ∈ R;
(ii) g : R2 → R2 , g(x, y) = (a + x, b + y), where a, b ∈ R?
8. Let a ∈ R and let f : R2 → R2 be defined by

f (x, y) = (x cos a − y sin a, x sin a + y cos a) .

Prove that f ∈ EndR (R2 ).

9. Determine the kernel and the image of the endomorphisms from Exercise 6.
10. Let V be a vector space over K and f ∈ EndK (V ). Show that the set

S = {x ∈ V | f (x) = x}

of fixed points of f is a subspace of V .

1
Seminar 3
1. Let M be a non-empty set and let SM = {f : M → M | f is bijective}. Show that
(SM , ◦) is a group, called the symmetric group of M .
2. Let M be a non-empty set and let (R, +, ·) be a ring. Define on RM = {f | f : M →
R} two operations by: ∀f, g ∈ RM ,

f +g : M → R, (f + g)(x) = f (x) + g(x) , ∀x ∈ M ,

f ·g : M → R, (f · g)(x) = f (x) · g(x) , ∀x ∈ M .

Show that (RM , +, ·) is a ring. If R is commutative or has identity, does RM have the same
property?
3. Prove that H = {z ∈ C | |z| = 1} is a subgroup of (C∗ , ·), but not of (C, +).
4. Let Un = {z ∈ C | z n = 1} (n ∈ N∗ ) be the set of n-th roots of unity. Prove that
Un is a subgroup of (C∗ , ·).
5. Let n ∈ N, n ≥ 2. Prove that:
(i) GLn (C) = {A ∈ Mn (C) | det(A) ̸= 0} is a stable subset of the monoid (Mn (C), ·);
(ii) (GLn (C), ·) is a group, called the general linear group of rank n;
(iii) SLn (C) = {A ∈ Mn (C) | det(A) = 1} is a subgroup of the group (GLn (C), ·).
6. Show that the following sets are subrings of the corresponding rings:
(i) Z[i] = {a+ bi |a, b ∈ Z} in (C, +, ·).
n a b o
(ii) M = a, b, c ∈ R in (M2 (R), +, ·).
0 c
7. (i) Let f : C∗ → R∗ be defined by f (z) = |z|. Show that f is a group homomorphism
between (C∗ , ·) and (R∗ , ·).  
∗ a b
(ii) Let g : C → GL2 (R) be defined by g(a + bi) = . Show that g is a group
−b a
homomorphism between (C∗ , ·) and (GL2 (R), ·).
8. Let n ∈ N, n ≥ 2. Prove that the groups (Zn , +) of residue classes modulo n and
(Un , ·) of n-th roots of unity are isomorphic.
a ∈ Z∗n .
9. Let n ∈ N, n ≥ 2. Consider the ring (Zn , +, ·) and let b
a is invertible ⇐⇒ (a, n) = 1.
(i) Prove that b
(ii) Deduce that (Zn , +, ·) is a field ⇐⇒ n is prime.
n  a b o
10. Let M = a, b ∈ R ⊆ M2 (R). Show that (M, +, ·) is a field isomorphic
−b a
to (C, +, ·).

1
Seminar 2
1. Let r, s, t, v be the homogeneous relations defined on the set M = {2, 3, 4, 5, 6} by
x r y ⇐⇒ x < y
x s y ⇐⇒ x|y
x t y ⇐⇒ g.c.d.(x, y) = 1
x v y ⇐⇒ x ≡ y (mod 3) .
Write the graphs R, S, T, V of the given relations.
2. Let A and B be sets with n and m elements respectively (m, n ∈ N∗ ). Determine the
number of:
(i) relations having the domain A and the codomain B;
(ii) homogeneous relations on A.
3. Give examples of relations having each one of the properties of reflexivity, transitivity
and symmetry, but not the others.
4. Which ones of the properties of reflexivity, transitivity and symmetry hold for the
following homogeneous relations: the strict inequality relations on R, the divisibility relation
on N and on Z, the perpendicularity relation of lines in space, the parallelism relation of
lines in space, the congruence of triangles in a plane, the similarity of triangles in a plane?
5. Let M = {1, 2, 3, 4}, let r1 , r2 be homogeneous relations on M and let π1 , π2 ,
where R1 = ∆M ∪ {(1, 2), (2, 1), (1, 3), (3, 1), (2, 3), (3, 2)}, R2 = ∆M ∪ {(1, 2), (1, 3)}, π1 =
{{1}, {2}, {3, 4}}, π2 = {{1}, {1, 2}, {3, 4}}.
(i) Are r1 , r2 equivalences on M ? If yes, write the corresponding partition.
(ii) Are π1 , π2 partitions on M ? If yes, write the corresponding equivalence relation.
6. Define on C the relations r and s by:
z1 r z2 ⇐⇒ |z1 | = |z2 | ; z1 s z2 ⇐⇒ arg z1 = arg z2 or z1 = z2 = 0 .
Prove that r and s are equivalence relations on C and determine the quotient sets (partitions)
C/r and C/s (geometric interpretation).
7. Let n ∈ N. Consider the relation ρn on Z, called the congruence modulo n, defined
by:
x ρn y ⇐⇒ n|(x − y) .
Prove that ρn is an equivalence relation on Z and determine the quotient set (partition)
Z/ρn . Discuss the cases n = 0 and n = 1.
8. Determine all equivalence relations and all partitions on the set M = {1, 2, 3}.
9. Let M = {0, 1, 2, 3} and let h = (Z, M, H) be a relation, where
H = {(x, y) ∈ Z × M | ∃z ∈ Z : x = 4z + y} .
Is h a function?
10. Consider the following homogeneous relations on N, defined by:
m r n ⇐⇒ ∃a ∈ N : m = 2a n ,
m s n ⇐⇒ (m = n or m = n2 or n = m2 ) .
Are r and s equivalence relations?

1
Seminar 6
1. Let v1 = (1, −1, 0), v2 = (2, 1, 1), v3 = (1, 5, 2) be vectors in the canonical real vector
space R3 . Prove that:
(i) v1 , v2 , v3 are linearly dependent and determine a dependence relationship.
(ii) v1 , v2 are linearly independent.
2. Prove that the following vectors are linearly independent:
(i) v1 = (1, 0, 2), v2 = (−1, 2, 1), v3 = (3, 1, 1) in R R3 .
(ii) v1 = (1, 2, 3, 4), v2 = (2, 3, 4, 1), v3 = (3, 4, 1, 2), v4 = (4, 1, 2, 3) in R R4 .
3. Let v1 = (1, a, 0), v2 = (a, 1, 1), v3 = (1, 0, a) be vectors in R R3 . Determine a ∈ R
such that the vectors v1 , v2 , v3 are linearly independent.
4. Let v1 = (1, −2, 0, −1), v2 = (2, 1, 1, 0), v3 = (0, a, 1, 2) be vectors in R R4 . Determine
a ∈ R such that the vectors v1 , v2 , v3 are linearly dependent.
5. Let v1 = (1, 1, 0), v2 = (−1, 0, 2), v3 = (1, 1, 1) be vectors in R R3 .
(i) Show that the list (v1 , v2 , v3 ) is a basis of the real vector space R3 .
(ii) Express the vectors of the canonical basis (e1 , e2 , e3 ) of R3 as a linear combination
of the vectors v1 , v2 and v3 .
(iii) Determine the coordinates of u = (1, −1, 2) in each of the two bases.
6. Let n ∈ N∗ . Show that the vectors

v1 = (1, . . . , 1, 1), v2 = (1, . . . , 1, 2), v3 = (1, . . . , 1, 2, 3), . . . , vn = (1, 2, . . . , n − 1, n)

form a basis of the real vector space Rn and write the coordinates of a vector (x1 , . . . , xn )
in this basis.          
1 0 0 1 0 0 0 0 1 0
7. Let E1 = , E2 = , E3 = , E4 = , A1 = ,
  0 0  0 0  1 0 0 1 0 1
1 1 1 1 1 1
A2 = , A3 = , A4 = . Prove that the lists (E1 , E2 , E3 , E4 ) and
0 0 1 0 1 1
(A1 , A
2 , A3 ,A4 ) are bases of the real vector space M2 (R) and determine the coordinates of
2 1
B= in each of the two bases.
1 0
8. Let R2 [X] = {f ∈ R[X] | degree(f ) ≤ 2}. Show that the lists E = (1, X, X 2 ),
B = (1, X − a, (X − a)2 ) (a ∈ R) are bases of the real vector space R2 [X] and determine the
coordinates of a polynomial f = a0 + a1 X + a2 X 2 ∈ R2 [X] in each basis.
9. Determine the number of bases of the vector space Z32 over Z2 .
10. Determine the number of elements of the general linear group (GL3 (Z2 ), ·) of in-
vertible 3 × 3-matrices over Z2 .

1
Seminar 1
1. Which ones of the usual symbols of addition, subtraction, multiplication and division
define an operation (composition law) on the numerical sets N, Z, Q, R, C?
2. Let A = {a1 , a2 , a3 }. Determine the number of:
(i) operations on A;
(ii) commutative operations on A;
(iii) operations on A with identity element.
Generalization for a set A with n elements (n ∈ N∗ ).
3. Decide which ones of the numerical sets N, Z, Q, R, C are groups together with the
usual addition or multiplication.
4. Let “ ∗ ” be the operation defined on R by x ∗ y = x + y + xy. Prove that:
(i) (R, ∗) is a commutative monoid.
(ii) The interval [−1, ∞) is a stable subset of (R, ∗).
5. Let “ ∗ ” be the operation defined on N by x ∗ y = g.c.d.(x, y).
(i) Prove that (N, ∗) is a commutative monoid.
(ii) Show that Dn = {x ∈ N | x/n} (n ∈ N∗ ) is a stable subset of (N, ∗) and (Dn , ∗) is a
commutative monoid.
(iii) Fill in the table of the operation “ ∗ ” on D6 .
6. Determine the finite stable subsets of (Z, ·).
7. Let (G, ·) be a group. Show that:
(i) G is abelian ⇐⇒ ∀x, y ∈ G, (xy)2 = x2 y 2 .
(ii) If x2 = 1 for every x ∈ G, then G is abelian.
8. Let “·” be an operation on a set A and let X, Y ⊆ A. Define an operation “∗” on the
power set P(A) by
X ∗ Y = {x · y | x ∈ X, y ∈ Y } .
Prove that:
(i) If (A, ·) is a monoid, then (P(A), ∗) is a monoid.
(ii) If (A, ·) is a group, then in general (P(A), ∗) is not a group.