finite-dimensional linear problem

finite-dimensional linear problem

Let

L

:

U

→

V

be a

linear mapping

, and let

v

∈

V

be given.
When both the domain

U

and

codomain

V

are

finite-dimensional

, a

linear equation

L

⁢

(

u

)

=

v

,

where

u

∈

U

is the

unknown

,
can be solved by means of

row reduction

. To do so, we need
to choose a basis

a

1

,

…

,

a

m

of the domain

U

, and a basis

b

1

,

…

,

b

n

of the codomain

V

. Let

M

be the

n

×

m

transformation matrix of

L

relative to these bases, and let

y

∈

ℝ

n

be the

coordinate vector

of

v

relative to the
basis of

V

. Expressing this in terms of matrix notation, we have

[

L

⁢

(

a

1

)

,

…

,

L

⁢

(

a

m

)

]

=

[

b

1

,

…

,

b

n

]

⁢

[

M

11

…

M

1

⁢

m

⋮

⋱

⋮

M

n

⁢

1

…

M

n

⁢

m

]

,

v

=

[

b

1

,

…

,

b

n

]

⁢

[

y

1

⋮

y

n

]

We can now restate the abstract linear equation as the matrix-vector
equation

M

⁢

x

=

y

,

with

x

∈

ℝ

m

unknown, or equivalently, as the following
system of

n

linear equations

M

11

⁢

x

1

+

⋯

+

M

1

⁢

m

⁢

x

m

=

y

1

⋮

⋱

⋮

⋮

M

n

⁢

1

⁢

x

1

+

⋯

+

M

n

⁢

m

⁢

x

m

=

y

n

with

x

1

,

…

,

x

m

unknown. Solutions

u

∈

U

of the abstract linear
equation

L

⁢

(

u

)

=

v

are in

one-to-one correspondence

with solutions of
the matrix-vector equation

M

⁢

x

=

y

. The correspondence is given by

u

=

[

a

1

,

…

,

a

m

]

⁢

[

x

1

⋮

x

m

]

.

Note that the dimension of the domain is the number of

variables

,
while the dimension of the codomain is the number of equations. The
equation is called

under-determined

or

over-determined

depending on
whether the former is greater than the latter, or vice versa. In
general, over-determined systems are

inconsistent

, while
under-determined ones have multiple solutions. However, this is a
“rule of thumb” only, and exceptions are not hard to find. A full
understanding of

consistency

, and multiple solutions relies on the
notions of kernel, image, rank, and is described by the

rank-nullity
theorem

.

Remark.

Elementary

applications exclusively on the
coefficient matrix and the right-hand vector, and neglect to mention
the underlying linear mapping. This is unfortunate, because the

concept

of a linear equation is much more general than the traditional
notion of “variables and equations”, and relies in an essential way
on the idea of a linear mapping. See the
example (

http://planetmath.org/UnderDeterminedPolynomialInterpolation

) on

polynomial

as a case in point. Polynomial interpolation
is a linear problem, but one that is specified abstractly, rather than
in terms of variables and equations.

Title

finite-dimensional linear problem

Canonical name

FinitedimensionalLinearProblem

Date of creation

2013-03-22 12:26:05

Last modified on

2013-03-22 12:26:05

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

12

Author

rmilson (146)

Entry type

Definition

Classification

msc 15A06

Related topic

LinearProblem

Related topic

RankNullityTheorem

Defines

system of linear equations

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