Ugrás a tartalomhoz

Convex Geometry

Csaba Vincze (2013)

University of Debrecen

2.4 Excercises

2.4 Excercises

Excercise 2.4.1 Check the affine dependence/independence of the system

v 1 = ( 1,2 , 3 ) , v 2 = ( 0,1 , - 1 ) , v 3 = ( 1,0 , 2 ) , v 4 = ( - 2,1 , 3 ) .

Excercise 2.4.2 Prove or disprove the following statements:

• Linearly independent systems are affinely independent.

• Affinely independent systems are linearly independent.

• Linearly dependent systems are affinely dependent.

• Affinely dependent systems are linearly dependent.

Excercise 2.4.3 Check the affine dependence/independence of the system

v 1 = ( 2,0 , - 1 ) , v 2 = ( 1,1 , 2 ) , v 3 = ( 0 , - 1,1 ) , v 4 = ( - 1,0 , 0 ) .

Excercise 2.4.4 Check the affine dependence/independence of the following systems of vectors.

v 1 = ( 1,0 , 0 , - 1 ) , v 2 = ( 2,1 , 1,0 ) , v 3 = ( 1,1 , 1,1 ) ,

v 4 = ( 1,2 , 3,4 ) , v 5 = ( 0,1 , 2,3 ) .

v 1 = ( 1,2 , 2 , - 1 ) , v 2 = ( 2,3 , 2,5 ) , v 3 = ( - 1,4 , 3 , - 1 ) ,

v 4 = ( 2,9 , 3,5 ) .

v 1 = ( - 3,1 , 5,3 , 2 ) , v 2 = ( 2,3 , 0,1 , 0 ) , v 3 = ( 1,2 , 3,2 , 1 ) ,

v 4 = ( 3 , - 5 , - 1 , - 3 , - 1 ) , v 5 = ( 3,0 , 1,0 , 0 ) .

Excercise 2.4.5 Find the affine coordinates of

( 2,1 ) , ( 1,1 ) , ( 1,1 / 3 )   a n d   ( 1,0 )

in the coordinate plane with respect to

v 1 = ( 2,0 ) , v 2 = ( 0,5 ) , v 3 = ( - 1,1 ) .

(2.14)

Using affine coordinates how to characterize points in the interior, points on the boundary or points outside of conv 2.14.

Excercise 2.4.6 Prove that

v 1 = ( 1 , - 1,2 , - 1 ) , v 2 = ( 2 , - 1,2 , 0 ) , v 3 = ( 1,0 , 2,0 ) ,

v 4 = ( 1,0 , 3,1 ) , v 5 = ( - 1,1 , 0,1 )

are affinely independent and find the affine coordinates of the origin in the coordinate space of dimension four.

Excercise 2.4.7 Consider the vector

v = 1 2 v 1 + 1 4 v 2 + 1 6 v 3 + 1 12 v 4 ,

where

v 1 = ( 1,0 ) , v 2 = ( 1,3 ) , v 3 = ( 4,3 ) , v 4 = ( 4,0 ) .

Use the procedure in the proof of Carathéodory's theorem 2.2.1 to reduce the number of the members in the convex combination as far as possible.

Excercise 2.4.8 Consider the vector

v = 1 5 v 1 + 1 5 v 2 + 1 5 v 3 + 1 5 v 4 + 1 5 v 5 ,

where

v 1 = ( 1,1 ) , v 2 = ( 4,1 ) , v 3 = ( 5,2 ) , v 4 = ( 2,3 ) , v 5 = ( 2,2 ) .

Use the procedure in the proof of Carathéodory's theorem 2.2.1 to reduce the number of the members in the convex combination as far as possible.

Excercise 2.4.9 Consider the vector

v = 1 10 v 1 + 3 20 v 2 + 1 4 v 3 + 1 5 v 4 + 2 25 v 5 + 11 50 v 6 ,

where

v 1 = ( 2,0 , - 1 ) , v 2 = ( 1,1 , 2 ) , v 3 = ( 0 , - 1,1 ) , v 4 = ( - 1,0 , 0 ) ,

v 5 = ( 1,0 , 1 ) , v 6 = ( 0 , - 3,3 ) .

Use the procedure in the proof of Carathéodory's theorem 2.2.1 to reduce the number of the members in the convex combination as far as possible.

Excercise 2.4.10 Consider the vector

v = 1 24 v 1 + 1 12 v 2 + 1 8 v 3 + 5 12 v 4 + 1 3 v 5 ,

where

v 1 = ( 2,0 , - 1 ) , v 2 = ( 1,1 , 2 ) , v 3 = ( 0 , - 1,1 ) , v 4 = ( - 1,0 , 0 ) ,

v 5 = ( 1,0 , 1 ) .

Use the procedure in the proof of Carathéodory's theorem 2.2.1 to reduce the number of the members in the convex combination as far as possible.