1. Lecture 14 Simplex, Hyper-Cube, Convex Hull and their Volumes
Shang-Hua Teng

2. Linear Combination and Subspaces in m-D
Linear combination of v1 (line)
{c v1 : c is a real number}
Linear combination of v1 and v2 (plane)
{c1 v1 + c2 v2 : c1 ,c2 are real numbers}
Linear combination of n vectors v1 , v2 ,…, vn
(n Space)
{c1v1 +c2v2+…+ cnvn : c1,c2 ,…,cn are real numbers}
Span(v1 , v2 ,…, vn)

3. Affine Combination in m-D

4. Convex Combination in m-Dp1 y p2 p3

5. Simplex
n dimensional simplex in m dimensions (n < m) is the set of all convex combinations of n + 1 affinely independent vectors

6. Parallelogram

7. Parallelogram

8. Hypercube
(1,1,1)
(0,1)
(0,0,1)
(1,0,0)
(1,0)
n-cube

9. Pseudo-Hypercube or Pseudo-Box
n-Pseudo-Hypercube
For any n affinely independent vectors

10. Convex Set

11. Non Convex Set

12. Convex Set A set is convex if the line-segment between
any two points in the set is also in the set

13. Non Convex Set A set is not convex if there exists a pair of points
whose line segment is not completely in the set

14. Convex Hull
Smallest convex set that contains all points

15. Convex Hull

16. Volume of Pseudo-Hypercube
n-Pseudo-Hypercube
For any n affinely independent vectors

17. Properties of Volume of n-D Pseudo-Hypercube in n-D

18. Signed Area and Volume volume( cube(p1,p2) ) = - volume( cube(p2,p1) )
(0,0) p1
volume( cube(p1,p2) ) = - volume( cube(p2,p1) )

19. Rule of Signed Volume n-D Pseudo-Hypercube in n-D

20. Determinant of Square Matrix
How to compute determinant or the volume of pseudo-cube?

21. Determinant in 2D Why? p2 =[b,d]T (0,0) p1 =[a,c]T
Invertible if and only if the determinant is not zero
if and only if the two columns are not linearly dependent

22. Determinant of Square Matrix
How to compute determinant or the volume of pseudo-cube?

23. Properties of Determinant
det I = 1
The determinant changes sign when sign when two rows are changed (sign reversal)
Determinant of permutation matrices are 1 or -1
The determinant is a linear function of each row separately
det [a1 , …,tai ,…, an] = t det [a1 , …,ai ,…, an]
det [a1 , …, ai + bi ,…, an] =
det [a1 , …,ai ,…, an] + det [a1 , …, bi ,…, an]
[Show the 2D geometric argument on the board]

24. Properties of Determinant and Algorithm for Computing it
[4] If two rows of A are equal, then det A = 0
Proof: det […, ai ,…, aj …] = - det […, aj ,…, ai …]
If ai = aj then
det […, ai ,…, aj …] = -det […, ai ,…, aj …]

25. Properties of Determinant and Algorithm for Computing it
[5] Subtracting a multiple of one row from another row leaves det A unchanged
det […, ai ,…, aj - t ai …] =
det […, ai ,…, aj …] + det […, ai ,…, - t ai …]
One can compute determinant by elimination
PA = LU then det A = det U

26. Properties of Determinant and Algorithm for Computing it
[6] A matrix with a row of zeros has det A = 0
[7] If A is triangular, then
det [A] = a11 a22 … ann
The determinant can be computed in O(n3)
time

27. Determinant and Inverse
[8] If A is singular then det A = 0. If A is invertible, then det A is not 0

28. Determinant and Matrix Product
[9] det AB = det A det B (|AB| = |A| |B|)
Proof: consider D(A) = |AB| / |B|
(Determinant of I) A = I, then D(A) = 1.
(Sign Reversal): When two rows of A are exchanged, so are the same two rows of AB. Therefore |AB| only changes sign, so is D(A)
(Linearity) when row 1 of A is multiplied by t, so is row 1 of AB. This multiplies |AB| by t and multiplies the ratio by t – as desired.