Enfoque metodológico

EXAMPLES

EXAMPLE6

Section 3.4 Special Subspaces for Systems and Mappings : Rank Theorem 153

Observe that in Example 4 and Exercise 3, the general solution is obtained from the solution space of the corresponding homogeneous problem (Example

1

^{and Exer­}

cise

1,

respectively) by a translation. We prove this result in the following theorem.

Let

jJ

be a solution of the system of linear equations

A

^x

= b, b

^*

0.

(

1)

^If

v

is any other solution of the same system, then

A(jJ -v) = 0,

^{so that}

jJ -v

is a solution of the corresponding homogeneous system

A

^x

= 0.

(2) If

h

is any solution of the corresponding system

Ax = 0,

^then

jJ + h

^{is a}

solution of the system

Ax = b.

Proof: (i) Suppose that

Av= b.

^Then

A(jJ - v) = AjJ -Av= b

^-

b = 0.

(ii) Suppose that

Ah= 0.

^Then

A(jJ + h) = AjJ +Ah= b + 0 = b.

•

The solution

jJ

of the non-homogeneous system is sometimes called a particular solution of the system. Theorem 2 can thus be restated as follows: any solution of the non-homogeneous system can be obtained by adding a solution of the corresponding homogeneous system to a particular solution.

Range of L and Columnspace of A

The range of a linear mapping L: JR11 � JRm is defined to be the set Range (L)

=

^{{L(x) E JRm I}

x

^{E JR11}}

Let ii

= [:]

and considec the lineac mapping prnj, ' R3 � R3. By definition, evecy image of this mapping is a multiple of

v,

so the range of the mapping is the set of all multiples of

v.

On the other hand, the range of perpil is the set of all vectors orthogonal to

v.

Note that in each of these cases, the range is a subset of the codomain.

If Lis a rotation, reflection, contraction, or dilation in JR3, then, because of the geome­

try of the mapping, it is easy to see that the range of Lis all of JR3.

EXAMPLE 7

EXERCISE4

Definition

Columnspace

EXAMPLE8

Let L : R2 � R3 be defined by L(x1, x2) = (2x1 - x2, 0, xi + x2). Find Range(L).

Solution: By definjtjon of the range, if L(x) is any vector in the range, then L(x) =

[

^2x1

^�

^x2

]

. Using vector operations, we can write this as

X1 + X2

This is focany x1,x2 e R, and so Range(L) =Span

{[�]

^·

nl}

^·

Let L : R3 � R2 be defined by L(x1, X2, X3) = (xi - x2, -2xi ⁺ 2x2 ⁺ x3). Find Range(L).

It is natural to ask whether the range of L can easily be described in terms of the matrix A of L. Observe that

L(x) =Ax=

[

^a1

_ctn]

^X1

_: ]

_{= X1G1 +}

"· + X11Gn Xn

Thus, the image of x under L is a linear combination of the columns of the matrix A.

The columnspace of an m x n matrix A is the set Col(A) defined by Col(A) = {Ax ^E R"' I x E R"J

Notice that our second interpretation of matrix-vector multiplication tells us that Ax is a linear combination of the columns of A. Thus, the columnspace of A is the set of all possible linear combinations of A. In particular, it is the subspace of R111 spanned by the columns of A. Moreover, if L : R11 � Rm is a linear mapping, then Range(L) = Col(A).

[

1 2

3] [

¹

Let A = 2 1 _ 1 _{and B =}

� -H

^Then

co1cAJ =span

{[ �]. [�]. [

_

;J}

_and coI(BJ =span

{[�]

^·

[-l]}

EXAMPLE9

EXERCISE 5

Theorem 3

EXAMPLE 10

Section 3.4 Special Subspaces for Systems and Mappings: Rank Theorem 155

If L is the mapping with standa<d matrix A =

[ � n

^then

Range(L) = Col(A) =Span

{[�]

^·

[i ]}

Find the standard matrix A of L(x1, x2, x3) ₌ (x1 - x2, -2x1 ₊ 2x2 ₊ x3) and show that Range(L) = Col(A).

The range of a linear mapping L with standard matrix A is also related to the system of equations Ax =

b

.

The system of equations Ax =

b

is consistent if and only if

b

is in the range of the linear mapping L with standard matrix A (or, equivalently, if and only if

b

is in the columnspace of A).

Proof: If there exists a vector x such that Ax =

b

, then

b

= Ax = L(x) and hence

b

_is

in the range of L. Similarly, if

b

is in the range of L, then there exists a vector x such

that

b

= L(x) =Ax. •

Suppose that L is a linear mapping with matrix A

[� n

Determine whether C ₌

u l

^and

^J

⁼

m

are in the range of L

Solution: c is in the range of L if and only if the system Ax = c is consistent. Sim­

ilarly,

J

is in the range of L if and only if Ax =

J

is consistent. Since the coefficient matrix is the same for the two systems, we can answer both questions simultaneously by row reducing the doubly augmented matrix

[

^A

I

^c

I ^J ]:

[

2 1 ^{1 1}

1 3 ^{-1 9}

� �]�[�

^{0 0}

^�

-] ^{1 2} 3

l

0 1

EXAMPLE 10

(continued)

Definition

Rows pace

EXAMPLE 11

Theorem 4

By considering the reduced matrix corresponding to

[

^A

I

^c

J

(ignore the last column), we see that the system Ax = c is consistent, so c is in the range of L. The reduced matrix corresponding to

[

^A

I ^J J

shows that Ax =

J

is inconsistent and hence

J

_is

not in the range of L.

Rowspace of ^A

The idea of the rowspace of a matrix A is similar to the idea of the columnspace.

Given an m x n matrix A, the rowspace of A is the subspace spanned by the rows of A (regarded as vectors) and is denoted Row(A).

[l

_{2 3}

] [1

³

]

Let A = 2

1

^_

1

^{and B =}

^�

^-

^�

^{. Then}

Row( A) =Span

m] .[_ m

^and Row(B) = Span

{ [ �]

.

[ _ i] . [ �])

To write a mathematical definition of the rowspace of A, we require linear com­

binations of the rows of A. But, matrix-vector multiplication only gives us a linear combination of the columns of a matrix. However, we recall that the transpose of a matrix turns rows into columns. Thus, we can precisely define the rowspace of A by

We now prove an important result about the rowspaces of row equivalent matrices.

If the m x n matrix A is row equivalent to the matrix B, then Row(A) = Row(B).

Proof: We will show that applying each of the three elementary row operations does not change the rows pace. Let the rows of A be denoted a

f

^{, ... , a}

�

and the rows of B be denoted by b1 CT , • • • ,

Suppose that B is obtained from A by interchanging two rows of A. Then, except for the order, the rows of A are the same as the rows of B; hence Row(A) = Row(B).

Suppose that B is obtained from A by multiplying the i-th row of A by a non-zero constant t. Then,

-+ -+ -+ -+

Row(B) = Span{b1, ... , bm} = Span{a1, ... , tai, ... ,

= {c1a1 + · · · + ci(tai) + · · · + cmam I ci E JR}

= {c1 a1 + ... + Cjtllj + ... + Cmllm I Cj E JR}

= Span{a1, • • • , am) = Row(A)

Theorem 5

EXAMPLE 12

Section 3.4 Special Subspaces for Systems and Mappings: Rank Theorem 157

Now, suppose that B is obtained from A by adding t times the i-th row to the }-th row. Then,

Row(B) ⁼ Span{b1, ...

, b111}

⁼ Span{a1,

.

.

. , aj +ta;, ... , a111}

=

{c1a1 + ... + c;a; + ... + Cj(aj +ta;)+ ... + Cmam IC;

E

JR}

=

{c1a1 +

^{· · ·}

+ (c; + cjl)a; +

^{· · ·}

+ cjaj +

^{· · ·}

+ cmam

^I

c;

E

JR}

= Span{a1,

..

^.

,a111}

⁼ Row(A)

By considering a sequence of elementary row operations, we see that row

equiva-lent matrices must have the same rowspace. •

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