23. Fundamental system of solutions of a homogeneous system of equations. Subspaces formed by solutions of a homogeneous linear system of equations.
Consider a homogeneous linear system of equations
Let
be a solution of the system. Write this solution as the vector
The collection of linearly independent solutions
of the system of equations (1) is called the fundamental
system of solutions
if any solution of the system of equations (1) can be represented in
form of linear combination of vectors
.
Theorem (on existence of fundamental system of solutions). If the rank of the matrix
is less than n then the system (1) has non-zero solutions. The number of vectors determining the fundamental system of solutions is found by the formula k = n – r where r is the rank of the matrix.
Thus, if we consider the linear space Rn of which vectors are all possible systems of n real numbers then the collection of all solutions of the system (1) is a subspace of the space Rn. The dimension of this subspace is equal to k.
24. Linear transformations. Examples of linear transformations. Actions over linear transformations.
Let
every element x
of a linear space
is corresponded a unique element
y
of a linear space
.
In this case we say an operator
A
is acting in
and having values in
with Ax
= y.
Operators are subdivided on mappings
if
and transformations
if
.
Furthermore we consider transformations acting in
.
We say that a transformation A is determined in a linear space if each vector x is corresponded the vector Ax by some rule. A transformation А is linear if for any vectors х and у and for any real number the following equalities hold:
A(x + y) = Ax + Ay, A( x) = Ax.
A linear transformation is called identity if it transforms each vector x to itself. An identity linear transformation is denoted by E. Thus, Ex = x.
Example.
Show
that the transformation
(where
is a real number) is linear.
Solution.
We
have
Thus, both conditions determining a linear transformation hold.
The considered transformation A is called a transformation of similarity.
Actions over linear transformations
Let A and B be arbitrary linear transformations in a linear space , λ be an arbitrary real number, and x be an element.
The
sum
of
linear transformations
A
and B
is called the transformation C1
determined
by the equality
.
Notation:
.
Lemma. The sum of two linear transformations is a linear transformation.
Proof:
Let
and
,
and let A
and B
be linear transformations,
.
Then
.
Zero transformation O is a transformation of a linear space V such that every element is corresponded the zero-element of this linear space.
A
transformation being opposite
for
a transformation A
is a transformation denoted by
such that every element
is corresponded the element
.
Obviously, zero and opposite transformations are linear.
The
product of a linear transformation A on number
λ is the transformation C2
determined by the equality
.
Notation:
.
Lemma.
The product of a linear transformation on number is a linear
transformation for which the following holds:
.
Theorem. The set of all linear transformations acting in a linear space is a linear space.
The
product of a linear transformation A on a linear transformation B
is called the transformation C3
determined by the equality
.
Notation:
.
Lemma. The product of linear transformations is a linear transformation.
Proof: Let A and B be linear transformations. Then
At
addition of linear transformations the commutative law holds, i.e.
.
The product AB
differs from the product BA
in general.
Let
A
and B
be linear transformations. The transformation
is called the commutator
of A
and B.
If A
and B
are commuting, i.e. AB
= BA,
then its commutator is zero transformation.