Download The Structure of Solution Sets to Linear Systems A system of m

The Structure of Solution Sets to Linear Systems
A system of m linear equations in n unknowns can
be put in the form of a matrix equation Ax = b
where A is an m × n matrix and b ∈ R m . Also, if
the system is consistent, any solution is a vector x
from R n .
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If n = 2, the individual equations are representable
geometrically as lines in the coordinate plane, and
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the solution set of the system has the form of either
a single point or an entire line. If n = 3, the
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equations are representable as lines or planes in
three-space, and the solution set of the system has
the form of either a single point, an entire line, or a
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whole plane. In fact, we have access to a deeper
understanding of the structure of these solutions.
As the example on pp. 21-22 illustrates, any
consistent system has a solution set with a certain
number of basic variables; the remaining variables
are free. The general solution to the system can
then be parametrized by expressing each basic
variable as the sum of some constant and a linear
combination of the free variables. This means that
the general solution can be expressed in vector
form as
x = p + x1 v 1 +  + x f v f ,
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where p is the vector containing these constants,
x 1 ,… x f are the free variables, and v 1 ,…, v f are
vectors containing the coefficents associated with
the free variables.
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Since the free variables are€just that – free, we can
arbitrarily assign them values to obtain different
solutions x to the original system. By setting all of
the free variables equal to 0, we notice that x = p
must be a solution to the original system. That is,
Ap = b. But then, if x represents any solution,
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A( x 1 v 1 +  + x f v f ) = A( x − p )
= Ax − Ap
= b− b
=0
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which shows that the system of equations
corresponding to the allied matrix equation Ax = 0,
which we call the homogeneous system with
coefficient matrix A, has as its solution set the span
of the vectors v 1 ,…, v f .
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One solution to the homogeneous system is clearly
the zero vector 0. It follows that the geometric
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characterization
of the homogeneous solution set is
that of a linear space, namely a point (which is
the origin itself) when there are no free variables
involved in the solution; a line, when there is
exactly one free variable; a plane, when there are
exactly two free variables; … All of these spaces lie
within R n and pass through its origin.
This characterization of solution sets is completely
€reversible: if v h is any solution to the homogeneous
system Ax = 0 and p is any particular solution to
the original system Ax = b, then x = p + v h must
be another
solution to Ax = b, since
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Ax = €
A( p + v h ) = Ap + Av h = b + 0 = b.
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This argument proves the
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Theorem If a linear system of equations, with
matrix equation Ax = b, is consistent and p is any
one particular solution, then every solution to the
system has the form x = p + v h where v h is a
solution €
to the associated homogeneous system
Ax = 0. //
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Geometrically, this says that the linear space
defining the solution set to Ax = b is obtained from
the linear solution space to the homogeneous
system Ax = 0 (which must pass through the origin
in R n ) by translating
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particular solution to the original system.
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Consequently, the two systems Ax = b and Ax = 0
have parallel solution sets (see Figure 5, p. 53).
In general, the solution set to the linear matrix
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equation Ax = b is parametrized in vector form as
follows:
(1) bring its augmented matrix to row reduced
echelon form;
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(2) solve for the basic variables in terms of the free
variables in the resulting equations; then
(3) translate these into vector notation, writing
the general solution vector x as a particular
solution vector p plus a linear combination
x 1 v 1 +  + x f v f of vector solutions to the
associated homogeneous system with the free
variables as parametric coefficients.
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