National University of Computer & Emerging Sciences
Islamabad
FAST School of Computing
Fall 2025
Islamabad Campus
MT1004 – Linear Algebra
Homework # 3
Linear Independence
1. Determine if the sets of vectors are linearly independent. For any sets that are linearly dependent,
find a dependence relationship among the vectors.
(a)
1
4
v1 =
3 ,
0
2
3
v2 =
4 ,
5
3
2
v3 =
2
4
0
u1 = 1 ,
2
2
u2 = 1 ,
3
2
u3 = 0
1
(b)
Subspaces
2. Determine whether W is a subspace of V .
x
(a) V = R3 , W = y : x = z − 3y + 1, z = − 32 y
z
x
(b) V = R3 , W = y ∈ R3 : x − 2y + z = 0, y + z = 0
z
W = { a + bx + cx2 : a + b + c = 0, }
(
!
)
a b
(d) V = M2×2 ,
W =
∈ M2×2 : c = b, d = 2a
c d
(c) V = P2 ,
3. If S1 and S2 are subspaces of a vector space V , then the union
S1 ∪ S2
is a subspace of V if and only if one subspace is contained in the other, i.e., S1 ⊆ S2 or S2 ⊆ S1 .
Consider the following subspaces of R3 :
(a) Let S1 = Span{(1, 2, 1)} (line in R3 ), and S2 = Span{(1, 5, −1), (2, 7, 0)} (plane in R3 ).
• Verify that line S1 in contained in the plane S2 .
• Conclude that S1 ∪ S2 is a subspace of R3 .
(b) Let S1 = Span{(1, 0, 0)} and S2 = Span{(1, 1, 0), (0, 1, 1)}.
• Show that neither S1 ⊆ S2 nor S2 ⊆ S1 .
• Conclude that S1 ∪ S2 is not a subspace of R3 .
1
0
