Brief review of linear algebra, plus some additional facts
1.
R
n
≡
x
=
x
1
.
.
.
x
n
±
±
±
±
±
±
±
x
i
real
.
2.
A linear combination
of vectors
x
1
···
,
x
k
in
R
n
is a vector of the form
x
=
c
1
x
1
+
+
c
k
x
k
where
c
1
, ..., c
k
are scalars.
3. A set of vectors
{
x
1
, ...,
x
k
}
is linearly dependent
if some nonzero linear combination
of them yields the zero vector; otherwise it is linearly independent
.
4. A subset
V
⊂
R
n
is a subspace
of
R
n
if
x
,
y
∈
V
=
⇒
c
x
+
d
y
∈
V
, i.e., if
V
is
closed under linear combination.
5. A set of vectors
{
x
1
, ...,
x
k
}
in a subspace
V
is a basis
for
V
if each
x
∈
V
can
be expressed uniquely as a linear combination of
x
1
, ...
x
k
. The dimension
of
V
, denoted
dim(
V
), is the number of vectors in a basis for
V
.
Result
: A set of vectors in
R
n
is a basis for
R
n
iﬀ there are
n
of them in all and they are
linearly independent.
6. span(
x
1
, ...,
x
k
)
is the set of all linear combinations of
x
1
, ...,
x
k
.
7.
R
m,n
≡
A
=
a
1
,
1
a
1
,n
.
.
.
.
.
.
a
m,
1
a
m,n
±
±
±
±
±
±
±
a
i,j
real
.
8. For
A
∈
R
m,n
...
column space(
A
)
≡
span(columns of
A
).
row space(
A
)
≡
span(rows of
A
).
range(
A
)
≡{
y
∈
R
m

y
=
A
x
for some
x
∈
R
n
}
.
null(
A
)
x
∈
R
n

A
x
=0
}
.
rank(
A
)
≡
dim(column space(
A
))
(= dim(row space (
A
))
9. The transpose
of
A
∈
R
m,n
is a matrix
A
T
∈
R
n,m
whose
i, j
th
entry is
a
j,i
.
10.
A
∈
R
n,n
is nonsingular
if there exists a matrix
A

1
∈
R
n,n
such that
AA

1
=
I
,
where
I
is the
n
×
n
identity matrix
.I
f
A

1
exists, it is also the case that
A

1
A
=
I
.
11.
Result
:(
AB
)
T
=
B
T
A
T
;(
AB
)

1
=
B

1
A

1
(assuming
A
and
B
are nonsingu
lar).
12. The inner product
(or dot product
, or scalar product
) of two vectors
x
,
y
∈
R
n
is:
(
x
,
y
)
≡
n
X
i
=1
x
i
y
i
(=
x
T
y
)
.
Typeset by
A
M
S
T
E
X