University of Fort Hare
Dept of Mathematics
MAT 223 Test
Marks: 60
27 October 2003
Time : 2 Hours
Question One
1.1 (a) Let V be the set of all real numbers. Define addition on V to be the
usual subtraction and scalar multiplication be the usual multiplication
of real numbers. So a ⊕ b = a − b and k a = ka. Is V a vector space
over R? Explain.
(2)
(b) Let V be the set of all real-valued functions on R. Define f ⊕ g by
(f ⊕ g)(x) = f (x) + g(x) and (k f )(x) = kf (x). What must be the
zero element and the multiplicative identity so that V is a real vector
space?
(2)
(c) Let W be a subset of a vector space V . Name the two properties
that must be satisfied for W to be a subspace of V .
(3)
1.2 Let 2 R2 be the set of all 2 × 2 matrices with real number entries. Define
addition and scalar multiplication on 2 R2 in the usual way.
" Let W #be
0 a12
.
the subset of 2 R2 consisting of matrices of the form A =
a21
0
(5)
(i) Show that W is a subspace of 2 R2 ; (ii) Find a basis for W .
1.3 Let W be the set of all singular 2 × 2 matices. Is W a subspace of 2 R2 ?
Explain.
(2)
1.4 Let W =
("
a b c
d 0 0
#
)
: c > 0 . Say why W is not a subspace of 2R3 .
(2)
1.5 (a) Define what is meant by a linearly (i) dependent set and (ii) independent set in a real vector space.
(3)
(b) Define what is meant by a set in a vector space V spans V .
(2)
[21]
1
Question Two
2.1 Let V = R3 (the vector space of all 1×3 matrices). Let S = {v1, v2, v3 , v4, v5 }
where v1 = [1 0 1]; v2 = [0 1 1]; v3 = [1 1 2]; v4 = [1 2 1]; v5 =
[−1 1 − 2]. Obtain a basis for V from S.
(8)
2.2 Let W be the subspace of R3 spanned by S = {[0 1 1]; [1 0 1]; [1 1 2]}.
Find dim W .
(3)
2.3 Let W = {[a a c d] : a, c, d ∈ R}. Find dim W as a subspace of R4 .









(3)




 .


1
1
 
0  ;  1
2.4 Let S =
Show that S is linearly independent. Let
0
0
W be the subspace of R3 spanned by S. Find a basis T of R3 containing
S.
(4)
2.5 Let S = {v1; · · · ; vn } be a basis for V . Show that T = {kv1; · · · ; kvn },
(k 6= 0), is a basis for V .
(3)
[21]
Question Three
3.1 Suppose V is an n-dimensional vector space and S = {v1; · · · ; vm } is a
set of m vectors in V . Show that if m < n, then S cannot span V .
(3)
3.2 
Find a
1 2

 2 3

 1 1


 3 5
2 3
basis
0 3
0 3
2 2
0 6
2 5

3.3 Let A =







for the
V of the homogeneous system
 solution

 space

1
x1
0




 x 
 0 
1 
 2 






1 
(6)
  x3  =  0  .




2   x4   0 
2
0
x5
1
2 3
2
1 4
−1 −1 2
0
1 2
1
1 1








. Find rank (A).
2
(5)
3.4 Use a matrix and elementary row operations
to
a basis
("
# find
"
# " for the
#
1 2
2 1
0 2
subsapace W of 2R2 spanned by S =
;
;
;
1 1
3 1
1 2
"
# "
#)
3 2
5
0
;
(5)
1 4
0 −1
[19]
END
3