Solutions for Selected problems of Homework 1
A. Agarwal
1016331
Linear Algebra1
Spring 200910
1.
Q22,P.14
Here is an
algebraic way
of finding
w
as a linear combination of
u
and
v
.
Linear combination
means, we have to find scalars
c
1
, c
2
such that
w
=
c
1
u
+
c
2
v
. The vector equation will look
like
"
2
9
#
=
c
1
"

2
3
#
+
c
2
"
2
1
#
This gives us
"
2
9
#
=
"

2
c
1
+ 2
c
2
3
c
1
+
c
2
#
Solving the system of equations will give us
c
1
= 2
, c
2
= 3.
2.
Q56(a),P.27
We will use the fact that

a

2
=
a
·
a

u
+
v

2
+

u

v

2
= (
u
+
v
)
·
(
u
+
v
) + (
u

v
)
·
(
u

v
)
= (
u
·
u
+
u
·
v
+
v
·
u
+
v
·
v
) + (
u
·
u

u
·
v

v
·
u
+
v
·
v
)
using the commutative property of the dot product
= (

u

2
+ 2
u
·
v
+

v

2
) + (

u

2

2
u
·
v
+

v

2
)
= 2(

u

2
+

v

2
)
3.
Q60,P.27
Use the fact

u
+
v

2
=

u

2
+ 2
u
·
v
+

v

2
that we established in the previous problem to
get

u
+
v

= 3.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
4.
If
k
v
k
= 5 and
k
w
k
= 3, what are the largest and smallest values of
k
v

w
k
? What
are the largest and smallest values of
v
·
w
?
Solu:
Let
θ
be the angle between
v
and
w
. Then
v
·
w
=
k
v
kk
w
k
cos
θ
= 15 cos
θ
. Using the same
idea as in previous problems.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 Dr.AnuragAgarwal
 Linear Algebra, Algebra, Scalar, Vector Space, Dot Product, Cos, solu

Click to edit the document details