144
C Linear or Vector Spaces
i.e., the equation
p
r=1 α r x r = 0 admits only the trivial solution. If this does not
hold, the vectors are said to be linearly dependent.
Note
1. A set of vectors x r , r = 1, . . . , p, of R n containing the zero vector is linearly
dependent since, if x p = 0,
p
r=1 α r x r = 0 holds for α 1 = · · · = α p−1 = 0 and
α p = 0.
2. If p > n, the set of vectors x r , r = 1, . . . , p, is linearly dependent. Let A
be an (n, p)-matrix whose columns are the vectors x r . The system of equations
A α =
p
r=1 α r x r = 0 has more variables than equations so that it admits nontrivial solutions, i.e.,
p
r=1 α r x r = 0 does not imply α r = 0 for all r = 1, . . . , p.
3. A direct consequence of the previous statement is that n is the maximum number
of linearly independent vectors in R n .
Example C.7 The unit vectors i, j, and k are linearly independent since
α 1 i + α 2 j + α 3 k =
⎡
⎣
α 1
α 2
α 3
⎤
⎦ = 0 if and only if α 1 = α 2 = α 3 = 0.
Similarly, the “vectors” cos(ν k t) and sin(ν k t) are linearly independent since
f (t) = 0 at all times t in [0, τ ], if and only if the coefficients {a k } and {b k } of
the representation in Eq. C.4 are zero.
Definition C.5 A subset B = {b 1 , · · · , b m } of a vector space V is a basis of V
if (1) the elements of every finite subset of B are linearly independent and (2) the
elements x of V can be represented uniquely by x =
m
r=1 α r b r , i.e., B spans the
vector space V.
Example C.8 The unit vectors i, j, and k of the physical space R 3 define a basis for
this space since they are linearly independent and they span the space. The latter
statement follows from the observation that i, j, k, and any vector x ∈ R 3 are
dependent, so that α
1 i+α
2 j+α
3 k+α
4 x = 0 holds with non-zero scalars, so that x
is a linear form of i, j, and k. The projections of x on the unit vectors are x, i, x, j,
and x, k so that this vector admits the representation x = =x, i i++x, j j++x, k k.
Example C.9 Similar considerations hold for the n-dimensional Euclidian space
R n . Its unit vectors and other linearly independent vectors, e.g., the eigenvectors
of an (n, n)-matrix a, define basis of this space.
Suppose first that a is an (n, n)-symmetric real-valued matrix, and denote by
x 1 , . . . , x n its eigenvectors. These eigenvectors and an arbitrary vector x of R n
are linearly dependent (see item 3 following Definition C.4) so that
n
r=1 α r x r +
α n+1 x = 0 is satisfied with non-zero scalars. Accordingly, x is a linear form of
{x r }. The scalars in the representation of x can be calculated uniquely by projection
as in R 3 .
Suppose now that the matrix a is not symmetric and denote by {u 1 , . . . , u n }
and {v 1 , . . . , v n } its right and left eigenvectors. The right vectors are linearly
C Linear or Vector Spaces
i.e., the equation
p
r=1 α r x r = 0 admits only the trivial solution. If this does not
hold, the vectors are said to be linearly dependent.
Note
1. A set of vectors x r , r = 1, . . . , p, of R n containing the zero vector is linearly
dependent since, if x p = 0,
p
r=1 α r x r = 0 holds for α 1 = · · · = α p−1 = 0 and
α p = 0.
2. If p > n, the set of vectors x r , r = 1, . . . , p, is linearly dependent. Let A
be an (n, p)-matrix whose columns are the vectors x r . The system of equations
A α =
p
r=1 α r x r = 0 has more variables than equations so that it admits nontrivial solutions, i.e.,
p
r=1 α r x r = 0 does not imply α r = 0 for all r = 1, . . . , p.
3. A direct consequence of the previous statement is that n is the maximum number
of linearly independent vectors in R n .
Example C.7 The unit vectors i, j, and k are linearly independent since
α 1 i + α 2 j + α 3 k =
⎡
⎣
α 1
α 2
α 3
⎤
⎦ = 0 if and only if α 1 = α 2 = α 3 = 0.
Similarly, the “vectors” cos(ν k t) and sin(ν k t) are linearly independent since
f (t) = 0 at all times t in [0, τ ], if and only if the coefficients {a k } and {b k } of
the representation in Eq. C.4 are zero.
Definition C.5 A subset B = {b 1 , · · · , b m } of a vector space V is a basis of V
if (1) the elements of every finite subset of B are linearly independent and (2) the
elements x of V can be represented uniquely by x =
m
r=1 α r b r , i.e., B spans the
vector space V.
Example C.8 The unit vectors i, j, and k of the physical space R 3 define a basis for
this space since they are linearly independent and they span the space. The latter
statement follows from the observation that i, j, k, and any vector x ∈ R 3 are
dependent, so that α
1 i+α
2 j+α
3 k+α
4 x = 0 holds with non-zero scalars, so that x
is a linear form of i, j, and k. The projections of x on the unit vectors are x, i, x, j,
and x, k so that this vector admits the representation x = =x, i i++x, j j++x, k k.
Example C.9 Similar considerations hold for the n-dimensional Euclidian space
R n . Its unit vectors and other linearly independent vectors, e.g., the eigenvectors
of an (n, n)-matrix a, define basis of this space.
Suppose first that a is an (n, n)-symmetric real-valued matrix, and denote by
x 1 , . . . , x n its eigenvectors. These eigenvectors and an arbitrary vector x of R n
are linearly dependent (see item 3 following Definition C.4) so that
n
r=1 α r x r +
α n+1 x = 0 is satisfied with non-zero scalars. Accordingly, x is a linear form of
{x r }. The scalars in the representation of x can be calculated uniquely by projection
as in R 3 .
Suppose now that the matrix a is not symmetric and denote by {u 1 , . . . , u n }
and {v 1 , . . . , v n } its right and left eigenvectors. The right vectors are linearly