Math 2270 Assignment 6
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Dylan Zwick
Fall 2012
Section 3.3 1, 3, 17, 19, 22
Section 3.4 1, 4, 5, 6, 18
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-
3.3 The Rank and Row Reduced Form
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3.3.1 Which of these rules gives a correct definition of the rank of A?
(a) The number of nonzero rows in R.
(b) The number of columns minus the total number of rows.
Jhe number of columns minus the number of free columns.
(d) The number of l’s in the matrix R.
04
i
(c)
1
c Qt
3.3.3 Find the reduced R for each of these (block) matrices:
(0 0 O
A=(003)
2 4 6)
A A
7
c=(AO
B=(AA)
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rc-l
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[c’a6
b
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3.3.17 (a) Suppose column j of B is a combination of previous columns of
B. Show that column j of AB is the same combination of previ
ous columns of AB. Then AB cannot have new pivot columns,
so rank(AB) <rank(B).
1 and A
2 so that rank(A
B)
1
(b) Find A
B=(
,
=
1 and rank A
B
2
=
0 for
).
A
-
-
(
Ab, t
-
-
(
--
b)
1
A
AL
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1(1
c{flk 0
7
/
3
I.
3.3.19 (Important) Suppose A and B are ii by n matrices, and AB
Prove from rank(AB) < rank(A) that the rank of A is n. So A is in
vertible and B must be its two-sided inverse (section 2.5). Therefore
BA I (which is not so obvious!).
=
rk
k (A )
fr (4)
rck(A)
Arc
k 4)
4
) [r ()Z
3.3.22 Express A and then B as the sum of two rank one matrices:
rank=2
4—
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(
/1
A=f 1
1 O’
14)
1 8)
I
B=
2 2
23
/0 0
l!O)4/o()
1)J
/0
0
5
‘e Complete Solution to Ax
=
b.
3.4.1 (Recommended) Execute the six steps of Worked Example 3.4 A to
describe the column space and nullspace of A and the complete so
lution Ax = b:
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A( 25761
1
/b
2
bf b
/L
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3 is the system solvable? Include
3.4.5 Under what condition on b
,b
1
,b
2
b as a fourth column in elimination. Find all solution when that
condition holds:
z
5
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Lf
+ 2y
+
41
+
5
y
y
9
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—
—
h(
-L
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x
2x
6
2z
=
4z
8z
=
62
=
63
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