Page 73 - DMTH502_LINEAR_ALGEBRA

Unit 2: Vector Subspaces

 Let S , S , ..., S are subsets of a vector space V, the set of all sums Notes
1 2 n
...
1 2 k
of vectors S is called the sum of the subsets ,S S ,...S and is denoted by
i i 1 2 k
k
S S ... S or by S .
1 2 k i
i=1

2.4 Keywords

Symmetric Matrix: An n × n matrix A over the field F is symmetric if A ij A ji , for each i and j. The
symmetric matrices form a subspace of the square of all n × n matrices over the field F.
Vector Subspace: Let V be a vector space over a field F. Then a non-empty subset W of V is called
a vector subspace of V if under the operations of V, W itself, is a vector space of F.

2.5 Review Questions

1. Consider the three sets A,B,C such that

A x ,x ;x x
1 2 1 2
B x 1 ,x 2 ;x x 0
1 2
C x ,x ;x x
1 2 1 2
which of these sets are subspace of V(2)? Give reasons.

y
x
y
2. Let V R 3 x , ,z ; , ,z R and Let W be the set of all triples (x, y, z) such that
2x – 3y + 4z = 0
Show that W is a subspace of V.

3. Let V be the vector space of functions from R into R let V be the subset of even functions
s
f x f x ; let V be the subset of odd functions f x f x . Then
0
(a) Prove that V and V are subspaces of V.
s 0
(b) Prove that V + V = V
s 0
(c) Prove that V V 0 null vector.
s 0
4. Let W and W be subspaces of a vector space V such that W 1 W 2 V and
1
2
W W 0 . Prove that for each in V there are unique vectors 1 in W and 2 in W
1 2 1 2
such that 1 2 .

LOVELY PROFESSIONAL UNIVERSITY 67