Homework 10
due
Friday
Dec 2, 10pm (or Friday in class)
Review of determinants and their properties
If
A
is a square
n
×
n
matrix, it represents a linear function
R
n
A
-→
R
n
(with the
same
R
n
as input and as output). The determinant of
A
is defined as the factor by which this
function multiplies volumes. Specifically, if
B
in
is some
n
-dimensional region in
R
n
, and
B
out
is the corresponding region obtained by applying
A
B
in
A
R
n
B
out
R
n
Then
det
A
=
±
volume(
B
out
)
volume(
B
in
)
. The
±
sign out front has to do with orientation, and will be
discussed momentarily. Also note that “volume” is interpreted in an
n
-dimensional sense; for
example in
R
1
“volume” means length, in
R
2
“volume” means area, in
R
3
“volume” means
volume, in
R
4
“volume” is a 4d volume, etc...
The definition of det
A
above does not depend on the choice of region
B
in
. The volume
of
any
region is multiplied by the same amount. For computations it’s convenient to choose
B
in
to be the unit cube (or square, or hypercube, depending on the dimension). The unit
cube has all its edges parallel to the standard basis vectors, and its volume is 1
B
in
= unit cube :
edges
1
0
.
.
.
0
,
0
1
.
.
.
0
,...,
0
0
.
.
.
1
,
volume=1
When
B
in
is the unit cube,
B
out
is a “parallelepiped” whose edges are parallel to the column
vectors of
A
. Thus if
A
=
(
v
1
v
2
· · ·
v
n
)
then
B
out
= parallelepiped:
edges
v
1
,
v
2
, ...,
v
n
Then we find that
det(
A
)
=
±
volume(parallelepiped with edges
v
1
,
v
2
, ...,
v
n
)
volume(unit cube)
=
±
volume(parallelepiped with edges
v
1
,
v
2
, ...,
v
n
)
.
For example, suppose we have a matrix
A
=
a
b
c
d
that encodes a function from
R
2
to
R
2
. This function maps the unit square to a parallelogram with edges determined by the
1