Math 2270 Assignment 7
-
Dylan Zwick
Fall 2012
Section 3.5 1, 2, 3, 20, 28
Section 3.6 1, 3, 5, 11, 24
-
-
1
3.5 Independence, Basis, and Dimension
-
,v
1
,v
2
,v
3
4 are dependent:
3 are independent but v
,v
1
,v
2
3.5.1 Show that v
)
)
v
=
3
1
lj
v=( 1
\0/
v’o0/
( VI *
0
=
4
v
3
4)
(2\
/
cI
liei7
(3
/
/0)
(
LQ
Cl V t
ç
V +
1i
(el)
501
flU(
i1
3
V
/
-
J1Y
40)
(-1)
-
bfe
±
°
-
I
(0
q
t
Vi
(- I) /
j
4
ide,idr
/
L/
1
V
3
2
q
I ieqi
e/9e4de,?1
\_/
p
)
Th
-
‘I
C
cç)
ca.
—
I
I)
If
—
—
0
—
ci
Li)
—
cç
If
cc
—
—
C
C
-
—
-.cDcD
—‘
I
—.-.
II
II
—‘
I
—.-‘
ii
I
-I
CD
CD
CD
C’)
0
CD
CD
(CD
3.5.3 Prove that if a
dependent:
=
0 or d
=
0 or
0 (3 cases), the columns of U are
f
b \
d e
0
U=(\O 0 f/
a
7
(lni
O
V
QC
15
17 0
I
/
0
dl
He
O)
re
co
f
oJ)
/e
+Jeii
i’j
€
oJ
o)
co)uninJ
4
. Then find a basis
3
3.5.20 Find a basis for the plane x 2y + 3z = 0 in R
for the intersection if that plane with the y plane. Then find a basis
for all vectors perpendicular to the plane.
—
1k e
p
H
(i
-
x
2
-
3) / X\
(
(
YI/
y)
ii
w
/
U1
yIl
/
iQ1)
(
C
he f/due
I
U
*J
7)
0L
5
(iT
3.5.28 Find a basis for the space of all 2 by 3 matrices whose columns add
to zero. Find a basis for the subspace whose rows also add to zero.
49
-c
/oo)
/obo
01/
C)/
/0
6
-h
/
(
3.6 Dimension of the Four Subspaces
-
3.6.1 (a) If a 7 by 9 matrix has rank 5, what are the dimensions of the four
subspaces? What is the sum of all four dimensions?
(b) If a 3 by 4 matrix has rank 3, whare are its column space and left
nulispace?
(I( (A119-$/
(/A
c+r I77
h)
(/4
o
I)
a
1
jj
1
LeF
7-],
J-iM
Aac
f
7
MaJ
ffrne,iii
0
\I
4-
\-$
[I
-f
r-
>
1
—
—7---
—
\1
—
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-c..
cç
I)
1
cç
‘I
11
-
0—_
CN
I
C
(D
C
(Th
0
ci)
(ID
3.6.5 If V is the subspace spanned by
1 )
f\1J
and
1
f\OJ
find a matrix
A that has V as its row space. Find a matrix B that has V as its
nulispace.
1)
(A
(I
1
L
/7 t 1
ce
So
(i
-
1)
9
(
x)
q
4
o)
e o
3.6.11 (Important) A is an rn by n matrix of rank r. Suppose there are right
sides b for which Ax b has no solution.
(a) What are all inequalities (< or
and r?
(b) How do you know that ATy
) that must be true between m, ri
0 has solutions other than y
A
=
0?
I)
T
(
yenef
I
e
•
A
rk,
AT
HM
c
d
k
(//
foer
frhe
T (M /e
A
+he
lce of A)
10
3.6.24 (Important) ATy d is solvable when d is in which of the four subf0
L
spaces? The solution y is unique when the j p
f2c(
contains only the zero vector.
A
-
.)
r\ ike
cQIvoJ2/e
i/re
11
of
11
d
‘