E3101: A study guide and review, Short Form Version
1.3
Marc Spiegelman
December 5, 2007
1
The Shortform
Okay, here is the entire study guide in compact form
Solving
A
x
=
b for
A
∈
R
n
×
n
•
Equation:
A
x
=
b
•
Algorithm: Gaussian Elimination
[
A
b
]
→
[
U
c
]
and backsubstitute
•
Factorization:
PA
=
LU
(or
PA
=
LDU
or
PA
=
LDL
T
if
A
T
=
A
. Also
A
=
CC
T
(Cholesky factorization for SPD matrices).
The Matrix Inverse
A

1
•
Definition:
AA

1
=
A

1
A
=
I
, solves
x
=
A

1
b
.
•
Existence:
A

1
exists iff Gaussian Elimination produces
n
pivots (i.e.
n
linearly inde
pendent columns).
•
Uniqueness: if
A
is invertible,
A

1
is unique and
x
=
A

1
b
is unique
•
Algorithm: GaussJordan Elimination
[
A I
]
→
[
U E
d
]
→
[
D E
u
E
d
]
→
[
I D

1
E
u
E
d
] =
[
I A

1
]
Product Rules
•
For Matrices
–
General
AB
: inner products must agree. In general
AB
6
=
BA
.
–
Inverse:
(
AB
)

1
=
B

1
A

1
(if
A
,
B
both square and invertible)
–
Transpose:
(
AB
)
T
=
B
T
A
T
(all
A
and
B
such that
AB
exists.
–
Determinant:

AB

=

A

B

1
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E3101: Study Guide 2007
2
•
For vectors
– x
T
y
: inner product, dot product, maps
R
n
→
R
– x
T
y
=
y
T
x
– x
T
x
=

x

2
, where

x

is length of
x
– xy
T
: outer product. Is a rank 1 matrix. if
x
∈
R
m
and
y
∈
R
n
then
xy
T
∈
R
m
×
n
– xy
T
6
=
yx
T
General Solutions of
A
x
=
b for
A
∈
R
m
×
n
•
Algorithm: GaussJordan Elimination
[
A
b
]
→
[
R
d
]
where
R
is
reduced row echelon
form
•
Identify rank of
A
(number of pivot columns) and label pivot and free columns.
•
Check
existence
of solution (
d
∈
C
(
R
)
implies
b
∈
C
(
A
)
).
•
Solve
R
x
p
=
d
for the particular solution
x
p
(i.e. combination of pivot columns and
no
free columns that add to
d
).
•
Find special solutions as basis of
N
(
R
) =
N
(
A
)
.
•
General solution is
x
=
x
p
+
N
c
if
b
∈
C
(
A
)
•
Note:
x
p
not usually entirely in
C
(
A
T
)
(i.e.
x
p
6
=
x
+
).
x
+
=
A
+
b
=
A
+
A
x
p
The four fundamental subspaces of
A
∈
R
m
×
n
•
Definition of Basis: a minimum set of linearly independent vectors that span a vector
space or subspace.
•
Definition of Dimension: the number of basis vectors for any subspace.
•
The four subspaces of a matrix
A
which is
m
×
n
with rank
r
Name
Symbol
Dimension
Basis
Row Space
C
(
A
T
)
⊂
R
n
r
linearly independent rows of
R
Null Space
N
(
A
)
⊂
R
n
n

r
special solutions of
A
x
=
0
Column Space
C
(
A
)
⊂
R
m
r
linearly independent columns of
A
Left Null Space
N
(
A
T
)
⊂
R
m
m

r
special solutions of
A
T
x
=
0
•
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 Spring '08
 Spiegelman
 Linear Algebra, projection matrices

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