Math 2270 Assignment 2
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Dylan Zwick
Fall 2012
Section 2.1 4,5,9,13,17
Section 2.2 3,6,11,12,19
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1
Section 2.1 Vectors and Linear Equations
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2.1.4 Find a point with z = 2 on the intersection line of the planes x + y +
3z
6 and r y + z = 4. Find the point with z = 0. Find a third
point halfway between.
—
L
ft)
3
X+y
y-L
Zi<
Xyo
1,
1o
LLLD
b ee
1
2.1.5 The first of these equations plus the second equals the third:
x + y + z =2
x + 2y + z = 3
2x + 3y + 2z = 5
The first two planes meet along a line. The third plane contains that
line, because if x, y, z satisfy the first two equations then they also
;cJ-ifi fi I,,cJ The equations have infinitely many solutions (the
whole l’ine L). Find three solutions on L.
y
2y
Xlyi
-J
Jit’KL
/20 ifl
r(
be Ie
po
olAet
2
e
2.1.9 Compute each Ax by dot products of the rows with the column vec
tor:
(a)
/
(
1 2 4\ /2
—2 3 1
2
—4 1 2
\\ 3
J
(A
L
(-
3 i)- (
/-X
(b)
(
C
3):
/5
-
/2 1 0 o\ /i\
12101(11
1
0 1 2 1
0 0 1 2J \2J
Jj
L1o)- ((1 i)
1 0 )- (ii
I
(0
5
( i)
(oo i)
A
I
:5
i+c1 :5
(I / I
-
L
(I I i)
—
3
2.1.13 (a) A matrix with rn rows and n columns multiplies a vector with
components to produce a vector with
t1
components.
11
(b) The planes from the m equations Ax = b are in
dimensional space. The combination of the columns of A is in
space.
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4
(x
2.1.17 Find the matrix P that multiplies
y )to give
I\zJ
matrix
Q that multiplies
z
)
to bring back
z ).
(\xJ
lxy
)
1K
Ieifr
5
Find the
Section 2.2 The Idea of Elimination
2
-
2.2.3 What multiple of equations 1 should be subtracted from equation 2?
2x
—x + 5y
=
—
=
6
0
After this elimination step, solve the triangular system. If the right
side changes to
( ),
—6
what is the new solution?
I
7—
TA,3
ecfo/
I
JO/
ye
If
Y
3
6
-lo
Jr1
2.2.6 Choose a coefficient b that makes this system singular. Then choose a
right side g that makes it solvable. Find two solutions in that singular
case,
2x +
4r +
by
8y
W
e
i1I
=
=
16
g
I5
/ I jfl
/d
Qff1//
711/
/
7
/(
O4iy
Ic
flo’e1
4
50/
5olFro4;
2 yo /
OIJ/
/
k
/1/
7
2.2.11 (Recommended) A system of linear equations can’t have exactly
1
two solutions. Why?
(a) If
(
(
Y
are two solutions, what is another solu
and
y
\zJ
ZJ
A
4j
x
tion?
VII
(C(()
$0
rctyh
(1))
()
J
e
I/ne Aaf
cd
(b) If 25 planes meet at two points, where else do they meet?
A eyy
oi
o4
IM Jie
1
Y
ou’re not begin asked to answer “why” here. Parts (a) and (b) lead you through an
explanation as to why.
8
eJ
7
2.2.12 Reduce this system to upper triangular form by two row operations
2x
+ 3y + z
4x + 7
y + 5z
2y + 2z
—
—
=
=
8
20
0
Circle the pivots. Solve by back substitution for z, y, r.
5rc
/oJ
ZJy
q
>1’
—7y
z
)
9
f
c°d
2.2.19 Which number q makes this system singular and which right side t
gives it infinitely many solutions? Find the solution that has z = 1.
2z
x + 4y
6z
x + 7y
3y + qz
=
1
6
=
t
—
—
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1_ -Frdr
2
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ir
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Y (n- (i) 6
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+
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