This preview shows pages 1–2. Sign up to view the full content.
Homework 2  Solutions
1.4 Problem 1
1. TRUE. Use the trivial linear combinations, taking all constants to be zero.
2. FALSE. By convention, span(
∅
)=
{
0
}
.
3. TRUE. See Theorem 1.5.
4. FALSE. Only multiplication by nonzero constants is allowed.
5. TRUE.
6. FALSE. See the system of equations in Example 2 of this section.
1.4 Problem 2b
See example 1 of this section.
1.4 Problem 4b
See example 2 of this section.
1.4 Problem 10
The span of
{
M
1
,M
2
3
}
consists of linear combinations of these matrices. LEt us consider an arbitrary
linear combination:
aM
1
+
bM
2
+
cM
3
=
±
ac
cb
²
Since this is the most general 2
×
2 symmetric matrix, we see that the span of these three matrices must be
all symmetric 2
×
2 matrices.
1.4 Problem 15
See class notes  this example was worked out in class.
1.5 Problem 1
1. FALSE. Consider the set
{
(1
,
0)
,
(1
,
1)
,
(2
,
2)
}
.
2. TRUE.
3. FALSE. This is a point of convention. If we assume
∅
to be independent, then taking the union of an
independent set
S
with the empty set will keep
S
∪∅
=
S
independent.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Winter '07
 Liu
 Linear Algebra, Algebra, Multiplication

Click to edit the document details