2/12/2002, LINEAR TRANSFORMATIONS (I) Math 21b, O. KnillHOMEWORK: due Thursday 2.1: 4,8*,10,20,28,42*, (Due next Tuesday: 2.2: 4,8,10,32,47*,50*, 2.3:10,20,26*,30,40,42*LINEAR TRANSFORMATION. A map T from Rnto Rmis called a linear transformation if there is am × n matrix with T~x = ~y.EXAMPLES. T(~x) = ~y · ~x from R3to R (A = ~y is a 1 × 3 matrix (row vector). T (x) = ~vx from R to R3.A = ~v is a 3 × 1 matrix (column vector). A~x = (x, y) from R3to R2, A is a 2 × 3 matrix.LINEAR TRANSFORMATIONS IN SPACE ACTING ON A BODY.PROPERTIES OF LINEAR TRANSFORMATIONS.T (~0) =~0 T (~x + ~y) = T (~x) + T (~y) T (λ~x) = λ~xIn words: Linear transformations are compatible with addition and scalar multiplication. It does not matter,whether we add two vectors before the transformation or add the transformed vectors.SHEAR:A =1 0−1 1A =1 −10 1SCALING:A =2 00 2A =1/2 00 1/2REFLECTION:A =cos(2α) sin(2α)sin(2α) − cos(2α)A =1 00 −1PROJECTION:A =1 00 0A =0 00 1ROTATION:A =−1 00 −1A =cos(α) sin(α)− sin(α) cos(α)BOOST:A =cosh(α) sinh(α)sinh(α) cosh(α)The boost is a basic Lorentz transformationin special relativity. It acts on vectors (x, ct),where t is time, c is the speed of light and x isspace.Unlike in Galileo transformation (x, t) 7→ (x + vt, t) (which is a shear), time t also changes during thetransformation. The transformation has the effect that it changes length (Lorentz contraction). The angle α isrelated to v by tanh(α) = v/c. One can write also A(x, ct) = ((x + vt)/γ, t + (v/c2)/γx), with γ =p1 − v2/c2.COLUMN VECTORS. A linear transformation T (x) = Ax with A =| | · · · |~v1~v2· · · ~vn| | · · · |has the propertythat the column vector ~v1, ~vi, ~vnare the images of the standard vectors ~e1=1···0. ~ei=0·1·0. ~en=0···1.QUIZ. Find the linear transformation which rotates a cube around the diagonal (1, 1, 1) by 120 degrees (2π/3).INVERSE OF A LINEAR TRANSFORMATION.If S is a transformation such that S(T ~x) = ~x, for every ~x, then S is called the inverse of T .SOLVING A LINEAR SYSTEM OF EQUATIONS. A~x =~b means to invert the linear transformation A. Ifthe linear system has exactly one solution, then an inverse is possible. We write ~x = A−1~b. The inverse of thelinear transformation is again a linear transformation.ON LINEAR TRANSFORMATIONS. Linear transformations generalize scaling x 7→ ax in one dimensions.They are important in geometry, art (i.e. perspective, coordinate transformations), CAD applications (i.e.rotations, scales), physics (i.e. Lorentz transformations), dynamics (linearisations of general maps are linearmaps), compression (Fourier transform), coding (many codes are linear codes).LINEAR TRANSFORMATION OR NOT? (The square to the right is the image of the square to the
View Full Document
Unlocking...