Understand Of Linear Algebra

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Understand Of Linear Algebra

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Understand Of Linear Algebra

  1. 1. Understand of linear algebra Edward J. Yoon (edward@udanax.org) April 18, 2008
  2. 2. What is the Matrix? • A matrix is a set of elements, organized into rows and columns rows a b  c d  columns  
  3. 3. Matrix Operations f  a  e b  f  a b   e c d    g   c  g d  h  Just add elements   h   f  a  e b  f  a b   e c d    g   c  g d  h  Just subtract elements   h   f  ae  bg af  bh a b   e Multiply each row   ce  dg c d   g cf  dh  by each column   h  
  4. 4. Vector Operations • Vector: 1 x N matrix a  • Interpretation: a line   in N dimensional v  b  space • Dot Product, Cross c   Product, and Magnitude defined y on vectors only v x
  5. 5. Vector Interpretation • Think of a vector as a line in 2D or 3D • Think of a matrix as a transformation on a line or set of lines  x  a b   x'  y   c d    y '    
  6. 6. Dot Product • Interpretation: the dot product measures to what degree two vectors are aligned A A+B = C (use the head-to-tail B method to combine vectors) C B A
  7. 7. Dot Product d  a  b  abT  a b c  e   ad  be  cf Think of the dot product  as a matrix multiplication f  The magnitude is the dot a  aaT  aa  bb  cc 2 product of a vector with itself The dot product is also related to a  b  a b cos( ) the angle between the two vectors – but it doesn’t tell us the angle
  8. 8. Cross Product • The cross product of vectors A and B is a vector C which is perpendicular to A and B • The magnitude of C is proportional to the cosine of the angle between A and B • The direction of C follows the right hand rule – this why we call it a “right-handed coordinate system” a  b  a b sin( )
  9. 9. Inverse of a Matrix • Identity matrix: AI = A • Some matrices have 1 0 0 an inverse, such that: 0 1 0  AA-1 = I I  • Inversion is tricky:  (ABC)-1 = C-1B-1A-1 0 0 1    Derived from non- commutativity property
  10. 10. Inverse of a Matrix 1. Append the identity matrix to A 2. Subtract multiples of the a b c 1 0 0 other rows from the first d e  f  0 1 0 row to reduce the  diagonal element to 1 g h i 0 0 1   3. Transform the identity matrix as you go 4. When the original matrix is the identity, the identity has become the inverse!
  11. 11. Determinant of a Matrix • Used for inversion a b  A • If det(A) = 0, then A c d   has no inverse • Can be found using det( A)  ad  bc factorials, pivots, and cofactors! 1  d  b 1 A • Lots of ad  bc  c a    interpretations – for more info, take 18.06
  12. 12. Determinant of a Matrix ab c f  aei  bfg  cdh  afh  bdi  ceg de gh i ab ca b ca b c Sum from left to right de fd e fd e f Subtract from right to left Note: N! terms gh ig h ig h i
  13. 13. Homogeneous Matrices • Problem: how to include translations in transformations (and do perspective transforms) • Solution: add an extra dimension 1  x  1   y x z 1   y  1  z     1  1
  14. 14. Ortho-normal Basis • Basis: a space is totally defined by a set of vectors – any point is a linear combination of the basis • Ortho-Normal: orthogonal + normal • Orthogonal: dot product is zero • Normal: magnitude is one • Example: X, Y, Z (but don’t have to be!)
  15. 15. Ortho-normal Basis x  1 0 0 x y  0 T y  0 1 0 xz  0 T z  0 0 1 yz  0 T X, Y, Z is an orthonormal basis. We can describe any 3D point as a linear combination of these vectors. How do we express any point as a combination of a new basis U, V, N, given X, Y, Z?
  16. 16. Ortho-normal Basis n1  a  u  b  u  c  u  a 0 0  u1 v1 0 b 0 u n2    a  v  b  v  c  v  v2   2    0 0 c   u3 n3  a  n  b  n  c  n     v3 (not an actual formula – just a way of thinking about it) To change a point from one coordinate system to another, compute the dot product of each coordinate row with each of the basis vectors.

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