Subspaces of Rn , Bases and Linear Independence
Definition. Consider vectors ~v1 , . . . , ~vr in Rn . An equation of the form c1~v1 + · · · + cr ~vr = ~0 is called a
linear relation among the vectors ~v1 , . . . , ~vr . If at least one of the ci is nonzero, then we call this a nontrivial
linear relation among ~v1 , . . . , ~vr .
 
 

2
0
1
7
1
 3
 
 

1. Let ~v1 = 
 1, ~v2 = 2, and ~v3 = 4.
5
7
−1

(a) Are there any nontrivial linear relations among these vectors? If so, find one.
(b) Are the vectors ~v1 , ~v2 , ~v3 linearly independent?
(c) Let V = span(~v1 , ~v2 , ~v3 ). Find a minimal set of vectors that span V . (We’ll call this a basis of
V .) How would you describe the shape of V ?
2. Give geometric descriptions of the following:
• 1 vector ~v1 in Rn is linearly independent ⇐⇒
.
• 2 vectors ~v1 , ~v2 in Rn are linearly independent ⇐⇒
.
• 3 vectors ~v1 , ~v2 , ~v3 in Rn are linearly independent ⇐⇒
.
• The span of 1 linearly independent vector in Rn is
.
• The span of 2 linearly independent vectors in Rn is
.
• The span of 3 linearly independent vectors in Rn is
.
• The smallest span of vectors is
.
1
x
x
2
3. Let V =
∈ R : x ≥ 0 . (In words, V is the set of vectors
in R2 with x ≥ 0.) Is V a subspace
y
y
of R2 ? Why or why not?
x
2
4. Let V =
∈ R : y = ±x . Is V a subspace of R2 ? Why or why not?
y
x
2
5. Let V =
∈ R : y = 3x + 1 . Is V a subspace of R2 ? Why or why not?
y
2
6. Decide whether each of the following planes in R3 is a subspace of R3 . If so, find a basis of the subspace.
 
x
(a) The plane consisting of all vectors y  satisfying 2x − 7y + 4z = 0
z
 
x
(b) The plane consisting of all vectors y  satisfying 2x − 7y + 4z = 1
z
3
7. Let A be an n × m matrix. Is im A a subspace of Rn ? Why or why not?
8. True or false: If ~v1 , . . . , ~v5 are linearly dependent vectors in R7 , then ~v5 must be in span(~v1 , . . . , ~v4 ).
9. If ~v1 , . . . , ~vr are vectors in Rm , show that span(~v1 , . . . , ~vr ) is closed under addition and scalar multiplication.
4