INTRODUCTION
TO
LINEAR
ALGEBRA
Fourth Edition
MANUAL FOR INSTRUCTORS
Gilbert Strang
Massachusetts Institute of Technology
math.mit.edu/linearalgebra
web.mit.edu/18.06
video lectures: ocw.mit.edu
math.mit.edu/
±
gs
www.wellesleycambridge.com
email: [email protected]
Wellesley  Cambridge Press
Box 812060
Wellesley, Massachusetts 02482
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2
Solutions to Exercises
Problem Set 1.1, page 8
1
The combinations give (a) a line in
R
3
(b) a plane in
R
3
(c) all of
R
3
.
2
v
C
w
D
.2; 3/
and
v
NUL
w
D
.6;
NUL
1/
will be the diagonals of the parallelogram with
v
and
w
as two sides going out from
.0; 0/
.
3
This problem gives the diagonals
v
C
w
and
v
NUL
w
of the parallelogram and asks for
the sides: The opposite of Problem 2. In this example
v
D
.3; 3/
and
w
D
.2;
NUL
2/
.
4
3
v
C
w
D
.7; 5/
and
c
v
C
d
w
D
.2c
C
d; c
C
2d/
.
5
u
C
v
D
.
NUL
2; 3; 1/
and
u
C
v
C
w
D
.0; 0; 0/
and
2
u
C
2
v
C
w
D
.
add first answers
/
D
.
NUL
2; 3; 1/
.
The vectors
u
;
v
;
w
are in the same plane because a combination gives
.0; 0; 0/
. Stated another way:
u
D NUL
v
NUL
w
is in the plane of
v
and
w
.
6
The components of every
c
v
C
d
w
add to zero.
c
D
3
and
d
D
9
give
.3; 3;
NUL
6/
.
7
The nine combinations
c.2; 1/
C
d.0; 1/
with
c
D
0; 1; 2
and
d
D
.0; 1; 2/
will lie on
a lattice. If we took all whole numbers
c
and
d
, the lattice would lie over the whole
plane.
8
The other diagonal is
v
NUL
w
(or else
w
NUL
v
). Adding diagonals gives
2
v
(or
2
w
).
9
The fourth corner can be
.4; 4/
or
.4; 0/
or
.
NUL
2; 2/
. Three possible parallelograms!
10
i
NUL
j
D
.1; 1; 0/
is in the base (
x

y
plane).
i
C
j
C
k
D
.1; 1; 1/
is the opposite corner
from
.0; 0; 0/
. Points in the cube have
0
²
x
²
1
,
0
²
y
²
1
,
0
²
z
²
1
.
11
Four more corners
.1; 1; 0/; .1; 0; 1/; .0; 1; 1/; .1; 1; 1/
. The center point is
.
1
2
;
1
2
;
1
2
/
.
Centers of faces are
.
1
2
;
1
2
; 0/; .
1
2
;
1
2
; 1/
and
.0;
1
2
;
1
2
/; .1;
1
2
;
1
2
/
and
.
1
2
; 0;
1
2
/; .
1
2
; 1;
1
2
/
.
12
A fourdimensional cube has
2
4
D
16
corners and
2
±
4
D
8
threedimensional faces
and
24
twodimensional faces and
32
edges in Worked Example
2.4 A
.
13
Sum
D
zero vector. Sum
D NUL
2
:
00
vector
D
8
:
00
vector.
2
:
00
is
30
ı
from horizontal
D
.
cos
±
6
;
sin
±
6
/
D
.
p
3=2; 1=2/
.
14
Moving the origin to
6
:
00
adds
j
D
.0; 1/
to every vector. So the sum of twelve vectors
changes from
0
to
12
j
D
.0; 12/
.
15
The point
3
4
v
C
1
4
w
is threefourths of the way to
v
starting from
w
.
The vector
1
4
v
C
1
4
w
is halfway to
u
D
1
2
v
C
1
2
w
. The vector
v
C
w
is
2
u
(the far corner of the
parallelogram).
16
All combinations with
c
C
d
D
1
are on the line that passes through
v
and
w
.
The point
V
D NUL
v
C
2
w
is on that line but it is beyond
w
.
17
All vectors
c
v
C
c
w
are on the line passing through
.0; 0/
and
u
D
1
2
v
C
1
2
w
. That
line continues out beyond
v
C
w
and back beyond
.0; 0/
. With
c
³
0
, half of this line
is removed, leaving a
ray
that starts at
.0; 0/
.
18
The combinations
c
v
C
d
w
with
0
²
c
²
1
and
0
²
d
²
1
fill the parallelogram
with
sides
v
and
w
. For example, if
v
D
.1; 0/
and
w
D
.0; 1/
then
c
v
C
d
w
fills the unit
square.
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