Principles of Mathematics in Operations Research

Solutions 215

2.3

1. Let n = 4 and characterize bases for the four fundamental subspaces
   related to A = [yi\y 2 \ • • • \yn]-

[A\\h

10 0-1

0 1 0-1

00-1 1

00 1-1

-10 0 1

1-10 0

0 1-10

0 0 1-1

1000

0 100

00 10

0001

->

10 0-1

0 1 0-1

01-1 0

00 1-1

-1 00 0

-1-10 0

0 010

0 00 1

1 000]

1-10 0

1 110

0 001

->

'100

0 10

001

000

-1

-1

-1

0

-1 0 00"

-1-1 00

-1 -1 -10

1111

=

r/ 3 |v^

0

Si]

SII\

where Vjv =

-1

-1

-1

Si = , Sn= [1111].

Thus, TZ(A) = Span{yi,y 2 ,y3}. Af(A) = Span{t}, where

-1 0 00

-1-1 00

-1 -1 -10

t =

-vN

h-3

1

1

1

1

Moreover,

K(AT) = Span <

-1

0

0

1

5

1

-1

0

0

J

0

1

-1

0

Span{-y 4 ,-y 1 ,-y 2 }

And finally, N(AT) = Span {Sn} = Span j [ 1 1 1 1 ]T\.

The case for n = 3 is illustrated in Figure S.4. y\ is on the plane defined

by Span{ei,e2}, y 2 is on the plane defined by Span {e 2 ,e^} and 2/3 is

on the Span{ei,es}. Let us take {2/1,2/2} in the basis for 11(A), which

defines the red plane on the right hand side of the figure. The normal to

the plane is defined by the basis vector of Af(A) = Span {[1,1,1]T}. We

have M(A) = {K{A))L since N{A) = tf(AT) (therefore, K(AT) = H(A)

by the Fundamental Theorem of Linear Algebra-Part 2) in this particular

exercise.

Let us discuss the general case. Let e = (!,-•• , 1)T

[A\\In] -»

ITI-I\VN

0

Si]

Sn