LECTURE14:
          EXAMPLESOFCHANGEOF
                       BASISAND
        MATRIXTRANSFORMATIONS.
               QUADRATICFORMS.
                      Prof. N. Harnew
                    University of Oxford
                          MT2012
     1
       Outline: 14. EXAMPLESOFCHANGEOFBASIS
               ANDMATRIXTRANSFORMATIONS.
                         QUADRATICFORMS.
      14.1 Examples of change of basis
          14.1.1 Representation of a 2D vector in a rotated coordinate
          frame
          14.1.2 Rotation of a coordinate system in 2D
      14.2 Rotation of a vector in fixed 3D coord. system
          14.2.1 Example 1
          14.2.2 Example 2
      14.3 MATRICESANDQUADRATICFORMS
          14.3.1 Example 1: a 2 × 2 quadratic form
          14.3.2 Example 2: another 2 × 2 quadratic form
          14.3.3 Example 3: a 3 × 3 quadratic form
      2
                  14.1 Examples of change of basis
      14.1.1 Representation of a 2D vector in a rotated coordinate
      frame
         ◮ Transformation of vector r from Cartesian axes (x,y) into frame
           (x′,y′), rotated by angle θ
      3
    x′ = r cosα                          y′ = r sinα
    x = r cos(θ +α)                      y = r sin(θ + α)
    → x′= xcosα                          → y′= ysinα
              cos(θ+α)                            sin(θ+α)
    x cosα = x′cosθcosα−x′sinθsinα       y sinα = y′ sinθcosα+y′cosθsinα
    Since x′sinα = y′cosα                Since y′cosα =x′sinα
          x = x′cosθ−y′sinθ                   y = x′sinθ+y′cosθ
   ◮ Coordinate transformation:                                                                                                                                           
           x      cosθ  −sinθ  x′        Theseequations                                                                                                             
                 =                    ′     (1)  relate the coordinates                                                                                                   
             y       sinθ   cosθ     y                                                                                                                                    
                                                                                                                                                                          
   ◮ Take the inverse:                         ofrmeasuredinthe                                                                                                           
                                               (x,y)framewith those                                                                                                       
                                                                                                                                                                    
              ′                                measuredintherotated                                                                                                       
             x        cosθ   sinθ     x                                                                                                                                   
       4     y′  =    −sinθ  cosθ     y     (2)  (x′,y′) frame                                                                                                            