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Understand Of Linear Algebra
Understand Of Linear Algebra
Understand Of Linear Algebra
Understand Of Linear Algebra
Understand Of Linear Algebra
Understand Of Linear Algebra
Understand Of Linear Algebra
Understand Of Linear Algebra
Understand Of Linear Algebra
Understand Of Linear Algebra
Understand Of Linear Algebra
Understand Of Linear Algebra
Understand Of Linear Algebra
Understand Of Linear Algebra
Understand Of Linear Algebra
Understand Of Linear Algebra
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Understand Of Linear Algebra

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Understand Of Linear Algebra

Understand Of Linear Algebra

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  • 1. Understand of linear algebra Edward J. Yoon (edward@udanax.org) April 18, 2008
  • 2. What is the Matrix? • A matrix is a set of elements, organized into rows and columns rows a b  c d  columns  
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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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