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Purdue University
MA 353: Linear Algebra II with Applications
Homework 3, due Feb. 5, solutions
(I)
Sec. 1.4
#13: Show that if
S
1
and
S
2
are subsets of a vector space
V
such
that
S
1
⊆
S
2
, then span
S
1
⊆
span
S
2
. In particular, if
S
1
⊆
S
2
and
span
S
1
=
V
, deduce that span
S
2
=
V
.
Proof.
Assume
S
1
⊆
S
2
. Let
x
∈
span
S
1
. Then
x
=
a
1
v
1
+
a
2
v
2
+
···
+
a
n
v
n
for some
v
i
∈
S
1
and
a
i
∈
F
. But since
S
1
⊆
S
2
, we know that
v
i
∈
S
2
.
Hence
x
∈
span
S
2
.
In particular, if span
S
1
=
V
, then
V
⊆
span
S
2
, which implies
span
S
2
=
V
.
±
Sec. 1.5
#1 (b): Any set containing the zero vector is linearly dependent.
Answer
: Yes.
Proof.
Let
S
=
{
0
V
,v
1
,v
2
,...,v
n
}
be a set containing the zero vector.
Then consider
1
·
0
V
+ 0
F
v
1
+ 0
F
v
2
+
···
+ 0
F
v
n
= 0
V
.
The left hand side is a nontrivial linear combination of the vectors in
S
, which equals the zero vector. Hence
S
is linearly dependent.
±
#9: Let
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This note was uploaded on 04/29/2010 for the course MA 353 taught by Professor Staff during the Spring '08 term at Purdue University.
 Spring '08
 Staff
 Linear Algebra, Algebra, Vector Space, Sets

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