lecture 14 simplex hyper cube convex hull and their volumes
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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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linear combination and subspaces in m d
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)

simplex
Simplex

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

hypercube
Hypercube

(1,1,1)

(0,1)

(0,0,1)

(1,0,0)

(1,0)

n-cube

pseudo hypercube or pseudo box
Pseudo-Hypercube or Pseudo-Box

n-Pseudo-Hypercube

For any n affinely independent vectors

convex set1
Convex Set

A set is convex if the line-segment between

any two points in the set is also in the set

non convex set1
Non Convex Set

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

whose line segment is not completely in the set

convex hull
Convex Hull

Smallest convex set that contains all points

volume of pseudo hypercube
Volume of Pseudo-Hypercube

n-Pseudo-Hypercube

For any n affinely independent vectors

signed area and volume
Signed Area and Volume

p2

(0,0)

p1

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

determinant of square matrix
Determinant of Square Matrix

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

determinant in 2d
Determinant in 2D

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

determinant of square matrix1
Determinant of Square Matrix

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

properties of determinant
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]
properties of determinant and algorithm for computing it
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 it1
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 it2
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

determinant and inverse
Determinant and Inverse
  • [8] If A is singular then det A = 0. If A is invertible, then det A is not 0
determinant and matrix product
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.
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