F2010, Homework 3.
Section 1.5.
14.
a) and b) Let us do it directly for an
m
×
n
matrix
A
and the unit vectors
e
j
.
Recall that
e
j
is a vector in
R
n
all the coordinates of which are 0, but for the
j
th
which
is equal to 1.
We use the help of the formula
c
i,j
=
∑
n
k
=1
a
i,k
b
k,j
to compute the
i
th
coordinate of
A
e
j
and we get (
A
e
j
)
i
=
∑
n
k
=1
a
i,k
b
k,j
=
a
i,j
. This precisely means that
A
e
j
=
a
j
,
j
th
column in the matrix
A
and this holds for
j
= 1
,...,n
.
c) If
A
x
=
0
for all vectors
x
∈
R
n
, it is true in particular for the
e
i
,i
= 1
,...,n
.
From question b) we have that all the
A
e
i
=
a
i
are
0
: all the columns of
A
are
0
which
precisely means that
A
=
O
.
17.
Using 14 b) we have:
AI
n
=
A
[
e
1
...
e
n
] = [
A
e
1
...A
e
n
] = [
a
1
...
a
n
] =
A
For
I
m
A
we go back to the formulas of page 59 and we remember that in
I
m
, the diagonal
elements
I
i,i
are equal to 1, the others
I
i,k
with
i
̸
=
k
are 0. The formula gives for the
product
c
i,j
= 1
×
a
i,j
+ 0
×
∑
k
̸
=
i
a
k,j
. Hence the result.
21)
It is important to understand the question:
AB
=
BA
only if
A
and
B
are
square matrices of the same size, means that (
AB
=
BA
) implies (“
A
and
B
are square
matrices of the same size”)
If you are not convinced, let us say the following. Formally, if
A
and
B
are not square
matrices of the same size, the equality
AB
=
BA
cannot hold that is:
“
A
and
B
are NOT square matrices of the same size” implies “NOT (
AB
=
BA
)” or by
considering the contrapositive:
NOT(NOT
AB
=
BA
)implies NOT( “
A
and
B
are NOT square matrices of the same
size”)
Nota: The statement we are asked to prove certainly does not mean that the equality holds
if
A
and
B
are square matrices of the same size which would be a false statement easy to
disprove by giving a counter example.
Let
A
be an
m
×
n
matrix, and
B
be a
p
×
q
matrix. For the equality
AB
=
BA
to
hold, both products
AB
and
BA
must exist and be of the same size.
We must have
n
=
p
for the existence of the product
AB
which will be of size
m
×
q
,
and
q
=
m
for the existence of the product
BA
which will be of size
p
×
n
.
If both products exist,
n
=
p
and
q
=
m
, they are square and of size
m
×
m
(actually
m
×
q
but
q
=
m
), and
n
×
n
(actually
p
×
n
but
n
=
p
) respectively.
If these products are equal they have the same size, so that
m
=
n
=
p
=
q
. The
matrices
A
and
B
are both square and they have the same size.
Although very simple, this exercise is already subtle enough, read carefully the so-