Geoscience Reference
In-Depth Information
Equivalent linear systems
—Two systems of linear equations in
n
unknowns are equivalent if
they have the same set of solutions.
Geometric multiplicity of an eigenvalue
—The geometric multiplicity of eigenvalue
c
of
matrix
A
is the dimension of the eigenspace of
c
.
Homogeneous linear system
—A system of linear equations
A
x
=
b
is homogeneous if
b
= 0.
Inconsistent linear system
—A system of linear equations is inconsistent if it has no solutions.
Inverse of a matrix
—Matrix
B
is an inverse for matrix
A
if
AB
=
BA
=
I
.
Invertible matrix
—A matrix is invertible if it has no inverse.
Least-squares solution of a linear system
—A least-squares solution to a system of linear
equations
A
x
=
b
is a vector
x
that minimizes the length of the vector
A
x
-
b
.
Linear combination of vectors
—Ve ctor
v
is a linear combination of the vectors
v
1
, …,
v
k
if
there exist scalars
a
1
, …,
a
k
such that
v
=
a
1
v
1
+ … +
a
k
v
k
.
Linear dependence relation for a set of vectors
—A linear dependence relation for the set of
vectors {
v
1
, …,
v
k
} is an equation of the form
a
1
v
1
+ … +
a
k
v
k
= 0, where the scalars
a
1
, …,
a
k
are zero.
Linearly dependent set of vectors
—The set of vectors {
v
1
, …,
v
k
} is linearly dependent if the
equation
a
1
v1
+ … +
a
k
v
k
= 0 has a solution where not all the scalars
a
1
, …,
a
k
are zero (i.e.,
if {
v
1
, …,
v
k
} satisfies a linear dependence relation).
Linearly independent set of vectors
—The set of vectors {
v
1
, …,
v
k
} is linearly independent if
the only solution to the equation
a
1
v
1
+ … +
a
k
v
k
= 0 is the solution where all the scalars
a
1
,
…,
a
k
are zero (i.e., if {
v
1
, …,
v
k
} does not satisfy any linear dependence relation).
Linear transformation
—A linear transformation from
V
to
W
is a function
T
from
V
to
W
such that
1.
T
(
u
+
v
) =
T
(
u
) +
T
(
v
) for all vectors
u
and
v
in
V
.
2.
T
(
a
v
) =
aT
(
v
) for all vectors
v
in
V
and all scalars
a
.
Nonsingular matrix
—Square matrix
A
is nonsingular if the only solution to the equation
A
x
= 0 is
x
= 0.
Null space of a matrix
—The null space of
m
×
n
matrix
A
is the set of all vectors
x
in
R
n
such
that
A
x
= 0.
Null space of a linear transformation
—The null space of linear transformation
T
is the set of
vectors
v
in its domain such that
T
(
v
) = 0.
Nullity of a linear transformation
—The nullity of linear transformation
T
is the dimension
of its null space.
Nullity of a matrix
—The dimension of its null space.
Orthogonal complement of a subspace
—The orthogonal complement of subspace
S
of R
n
is
the set of all vectors
v
in R
n
such that
v
is orthogonal to every vector in
S
.
Orthogonal set of vectors
—A set of vectors in R
n
is orthogonal if the dot product of any two
of them is 0.
Orthogonal matrix
—Matrix
A
is orthogonal if
A
is invertible and its inverse equals its trans-
pose; that is,
A
-1
=
A
T
.
Orthogonal linear transformation
—Linear transformation
T
from
V
to
W
is orthogonal if
T
(
v
) has the same length as
v
for all vectors
v
in
V
.
Orthonormal set of vectors
—A set of vectors in R
n
is orthonormal if it is an orthogonal set
and each vector has length 1.
Range of a linear transformation
—The range of linear transformation
T
is the set of all vec-
tors
T
(
v
), where
v
is any vector in its domain.
Rank of a matrix
—The rank of matrix
A
is the number of nonzero rows in the reduced row
echelon form of
A
; that is, the dimension of the row space of
A
.
Rank of a linear transformation
—The rank of a linear transformation (and hence of any
matrix regarded as a linear transformation) is the dimension of its range. Note that a theo-
rem tells us that the two definitions of rank of a matrix are equivalent.
Search WWH ::
Custom Search