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# Lecture 14 Simplex, Hyper-Cube, Convex Hull and their Volumes - PowerPoint PPT Presentation

Lecture 14 Simplex, Hyper-Cube, Convex Hull and their Volumes. Shang-Hua Teng. Linear Combination and Subspaces in m-D. Linear combination of v 1 (line) { c v 1 : c is a real number} Linear combination of v 1 and v 2 (plane) { c 1 v 1 + c 2 v 2 : c 1 ,c 2 are real numbers}

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### Lecture 14Simplex, Hyper-Cube, Convex Hull and their Volumes

Shang-Hua Teng

Linear Combination and Subspaces in m-D

• Linear combination of v1(line)

{c v1: c is a real number}

• Linear combination of v1and v2(plane)

{c1v1 + c2v2: c1 ,c2are real numbers}

• Linear combination of n vectors v1 , v2 ,…, vn

(n Space)

{c1v1 +c2v2+…+ cnvn: c1,c2 ,…,cnare real numbers}

Span(v1 , v2 ,…, vn)

p1

y

p2

p3

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

(1,1,1)

(0,1)

(0,0,1)

(1,0,0)

(1,0)

n-cube

n-Pseudo-Hypercube

For any n affinely independent vectors

A set is convex if the line-segment between

any two points in the set is also in the set

A set is not convex if there exists a pair of points

whose line segment is not completely in the set

Smallest convex set that contains all points

n-Pseudo-Hypercube

For any n affinely independent vectors

p2

(0,0)

p1

volume( cube(p1,p2) ) = - volume( cube(p1,p2) )

How to compute determinant or the volume of pseudo-cube?

p2 =[b,d]T

Why?

(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

How to compute determinant or the volume of pseudo-cube?

• 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]

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 a= aj then

• det […, ai ,…, aj …] = -det […, ai ,…, aj …]

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 - tai …] =

det […, ai ,…, aj …] + det […, ai ,…, - tai …]

• One can compute determinant by elimination

• PA = LU then det A = det U

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

• [8] If A is singular then det A = 0. If A is invertible, then det A is not 0

• [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.