Math 2270 Assignment 8
-
Dylan Zwick
Fall 2012
Section 4.1 6, 7, 9, 21, 24
Section 4.2 1, 11, 12, 13, 17
-
-
1
4.1 Orthogonality of the Four Subspaces
-
4.1.6 The system of equations Ax
=
b has no solution (they lead to 0
x + 2y + 2z
2x + 2y + 3z
3x + 4y + 5z
=
=
=
1):
5
5
9
Find numbers Yi, Y2, y3 to multiply the equations so they add toO = 1.
You have found a vector y in which subspace? Its dot product yTb is
1, so no solution x.
5
50)
t5
7
y
Yi
-17
2
4.1.7 Every system with no solution is like the one in Problem 4.1.6. There
are numbers Yh.
Ym that multiply the ‘in equations so they add up
to 0 = 1. This is called Frendholm’s Alternative:
Exactly one of these problems has a solution
Ax
=
b
ATy
OR
0
with
yTb
=
1.
If b is not in the column space of A, it is not orthogonal to the nullspace
=
1 and
= 1 and x
2
of AT. Multiply the equations x
= 1 by numbers yi, Y2, y3 chosen so that the equations add up
to 0 = 1.
—
—
—
1 -xi
(x
(y -y,)
Hi-I
YL
I
(H)
3
-
1
(
-xi
4.1.9 If ATAx = 0 then Ax = 0. Reason: Ax is in the nulispace of AT and
7 oycl1( 1.
also in the c lam
ae of A and those spaces are cvii
ATA
has the same nulls pace as A. This key fact is repeated in
Conclusion:
the next section.
4
Suppose S is spanned by the vectors (1,2,2,3) and (1,3,3,2). Find
two vectors that span S. This is the same as solving Ax = 0 for
which A?
A
/1
(
x I
t
4
)ci
I
(9)
(A)
5
7
4.1.24 Suppose an n by n matrix is invertible: AA’ = I. Then the first
column of A’ is orthogonal to the space spanned by which rows of
A?
C
YO)
1)
o1
y
-k
n
rd
6
4.2 Projections
-
4.2.1 Project the vector b onto the line through a. Check that e is perpen
dicular to a:
1
and
a=(1
1J
2
(a)b=2)
(b) b
=
(
3)
b
and
(42 +
H H I
-
a
=
—1 \
(_3
—1 j
).
9
5
(
e
5J-
h)
—5
-iio
-
ffj
-
e
/\
(/;51)
/
o
(j
7
i
4.2.11 Project b onto the column space of A by solving ATA*
p=Ax:
ATb and
=
(1
(a)A
oi)
o 0
=
(b)A
=
(1 i
ii)
o i
Find e
b
=
At( 1
—
and
b=(3)
4j
and
b=(\4)
6J
p. It should be perpendicular to the columns of A.
° °
o
1
AA Hto
/
HO)
/
(H
IJH
; (&J
(i
00)
1 i
/
3)
()
If) ()
/
/
/ /
I
II
AT
b
—
I /
(0
x1
L)
/x,1
7-
yz5
II o)I
I I
IL)
)(t)
/0
I& j::: 0
L
/
t
A
5
(oJ(3)
(t(o/\
-
()
XI
(
-
/-
(
8
50
o
I
fe
(ofmS
o)
I I
(‘
(I)
(1\
poferf
61/
.
L4Ji
17
2Compute the projection matrices Pi and P
2 onto the column spaces
Verify
that
b
1
P
gives
Problem 4.2.11.
the first projection p
. Also
1
verify P = P
.
2
P,
A (AAi’A
AA
(oo
(4) (
AA)
(oô
-
o
()
—
1
[ioo /z
O(Oj/
oo/(9
/
(
(l
1
O
I
(co
oo
/
/3
I z z
H
LI
LL)
)(i) (L1)(iio)
HO
-1
(
•iO
z
/1
‘z-o
Gal)
(/1±)
1
r
4.2.13 (Quick and Recommended) Suppose A is the 4 by 4 identity matrix
with its last column removed. A is 4 by 3. Project b = (1, 2, 3, 4) onto
the column space of A. What shape is the projection matrix P and
what is P?
/
/4
OIQ
Oci
(CUe)
ft
o
ID
Cod
/ (ô
JIO(Oj
T
A
iJ
z
1
(AtA)
/ ooo)
PA (AA)A
(H
O(o
oo
/
C)iQO
/
I k
-
(Oo / i
ClOd
(0 Ii 3
I!
\
i
/
Oo
OIc
/
J
/?)
/ 3/
(0/
10
0o
/9
IC
cr
o,
(‘1
U
H
.1
H
H
(I
1
H
(I
*
H
H
H
Ic
H
lx
/\
—
c:z