Department of Computing, Statistical and Mathematical Sciences, University of Namibia,
Windhoek
S3511ML Linear Algebra I Tutorial 4 29 April 2024
Question 1. Define the following. Make use of your lecture’s notes to understand and be able to give
clear definitions.
(a) A field F.
(b) A vector space Vover the field F.
(c) A family of elements of a set S.
(d) A subspaces Sof a vector space V.
[(e) T+Sand T⊕S, where Tand Sare subspaces of a vector space V.
(f) T∪Sand T∩S, where Tand Sare subspaces of a vector space V.
(g) The spanning set of vectors.
Question 2 Prove the following statements. Please try to understand more proving of statements
given in your lecture’s notes.
(a) The set of rational numbers Qis a field
(b) The set of Galois 2 is a field.
(c) The set {0}is a subspace of the vector space Vand 0belongs to every subspace of V.
(d) T∩Sis a subspace whenever Tand Sare subspaces of a vector space V.
Question 3 Disprove the following statements (with an example).
(a) The set of Nis not a vector space.
(b) If Tand Sare subspaces of a vector space V, then S∪Tis not a subspace of V.
Question 4
1. Determine whether or not, the given vectors (v1, v2, v3) are in the spanning set of the vector b, in
each case
(a) v1=
2
1
4
, v2=
−1
−2
1
, v3=
3
3
−1
and b=
10
11
4
.
(b) v1=
2
−1
3
, v2=
4
2
1
and b=
2
−5
8
.
(c) v1=
2
−1
3
, v2=
4
2
1
and b=
−2
9
0
.
1
