Homework due Class 39
1. Let
A
be the following matrix:
[ 1
2
3 ]
[ 3
4
5 ]
[ 2
1
3 ]
[ 3
3
6 ]
[ 4
6
8 ]
Find a vector
v=(x,y,z)
which makes
Av
closest to the vector (1,2,3,4,5). Thus, we want to opimize the following:
[ 1
2
3 ]
[ 3
4
5 ]
[ 2
1
3 ]
[ 3
3
6 ]
[ 4
6
8 ]
*
[
x
]
[
y
]
[
z
]
=
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
From the webnotes, (and from class), we can work this problem by letting:
c
1
=
(1,3,2,3,4)
c
2
=
(2,4,1,3,6)
c
3
=
(3,5,3,6,8)
and finding the projection of (1,2,3,4,5) on {
c
1
, c
2
, and
c
3
}.
First, we will use the Gramm Schmidt Algorithm to find an orthogonal basis of the space spanned by {
c
1
, c
2
, and
c
3
}.
v
1
=
c
1
= (1,3,2,3,4)
v
2
=
c
2
- [(
c
2
,
v
1
)/(
v
1
,
v
1
)]*
v
1
= (2,4,1,3,6) - 49/39 * (1,3,2,3,4) = (29/39, 9/39, -59/39, -30/39, 38/39)
Now, we will define a new
v
2
which is 39*(the old
v
2
) thus, the new
v
2
= (29,9,-59,-30,38)
v
3
=
c
3
- [(
c
3
,
v
1
)/(
v
1
,
v
1
)]*
v
1
- [(
c
3
,
v
2
)/(
v
2
,
v
2
)]*
v
2
= (3,5,3,6,8) - 74/39 * (1,3,2,3,4) - 79/6747 * (29,9,-59,-30,38)
= (132/173, -138/173, -18/173, 114/173, -6/173)
Now, we will define a new
v
3
which is 173*(the old
v
3
) thus, the new
v
3
= (132, -138, -18, 114, -6)
Thus,
v
1
=
c
1
v
2
= 39*
c
2
- 49*
c
1
v
3
= 173*
c
3
- 12802/39*
c
1
- 79/39*(39*
c
2
- 49*
c
1
)
= 173*
c
3
- 79*
c
2
-229*
c
1
Now, letting
b
= (1,2,3,4,5), we will find the proj(
b
).
proj(
b
) = [(
b,v
1
)/(
v
1
,v
1
)]*
v
1
+ [(
b,v
2
)/(
v
2
,v
2
)]*
v
2
+ [(
b,v
3
)/(
v
3
,v
3
)]*
v
3
= 45/39*
v
1
+ (-60)/6747*
v
2
+ 228/49824*
v
3
We need to represent this as
x*c
1
+ y*c
2
+ z*c
3
Thus, we have:
proj(
b
) = 45/39*
c
1
+ (-60)/6747*(39*
c
2
- 49*
c
1
) + 228/49824*(173*
c
3
- 79*
c
2
-229*
c
1
)
= 45/39*
c
1
- 2340/6747*
c
2
+ 2940/6747*
c
1
+ 39444/49824*
c
3
- 18012/49824*
c
2
- 52212/49824*
c
1
= 13/24*
c
1
- 17/24
c
2
+ 19/24*
c
3
Thus:
x
=
13/24
y
=
-17/24
z
=
19/24