geometry1geometry210/6/2003, LINEAR TRAFOS IN GEOMETRY Math 21b, O. KnillHW: section 2.2: 6,8,16,30,34,47*,50*LINEAR TRANSFORMATIONS DE-FORMING A BODYA CHARACTERIZATION OF LINEAR TRANSFORMATIONS: a transformation T from Rnto RmwhichsatisfiesT (~0) =~0, T (~x + ~y) = T (~x) + T (~y) and T (λ~x) = λT (~x)is a linear transformation.Proof. Call ~vi= T (~ei) and define S(~x) = A~x. Then S(~ei) = T (~ei). With ~x = x1~e1+ ... + xn~en, we haveT (~x) = T (x1~e1+ ... + xn~en) = x1~v1+ ... + xn~vnas well as S(~x) = A(x1~e1+ ... + xn~en) = x1~v1+ ... + xn~vnproving T(~x) = S(~x) = A~x.SHEAR:A =1 01 1A =1 10 1In general, shears are transformation in the plane with the property that there is a vector ~w such that T ( ~w) = ~wand T (~x) − ~x is a multiple of ~w for all ~x. If ~u is orthogonal to ~w, then T (~x) = ~x + (~u · ~x) ~w.SCALING:A =2 00 2A =1/2 00 1/2One can also look at transformations which scale x differently then y and where A is a diagonal matrix.REFLECTION:A =cos(2α) sin(2α)sin(2α) − cos(2α)A =1 00 −1Any reflection at a line has the fo rm of the matrix to the left. A r e flec tio n a t a line containing a unit vector ~uis T (~x) = 2(~x · ~u)~u − ~x with matrix A =2u21− 1 2u1u22u1u22u22− 1PROJECTION:A =1 00 0A =0 00 1A projection onto a line containing unit vector ~u is T (~x) = (~x · ~u)~u with matrix A =u1u1u2u1u1u2u2u2ROTATION:A =−1 00 −1A =cos(α) − sin(α)sin(α) cos(α)Any rotation has the form of the matrix to the right.ROTATION-DILATION:A =2 −33 2A =a −bb aA rotation dilation is a composition of a rotation by angle arctan(y/x) and a dilation by a factorpx2+ y2. Ifz = x + iy and w = a + ib and T (x, y) = (X , Y ), then X + iY = z w. So a rotation dilation is tied to the processof the multiplication with a complex number.BOOST:A =cosh(α) sinh(α)sinh(α) cosh(α)The boost is a basic Lorentz transformationin s pecial 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 transformatio n 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, c t) = ((x + vt)/γ, t + (v/c2)/γx), with γ =p1 − v2/c2.ROTATION IN SPACE. Rotations in space are defined by an axes of rotationand an angle. A rotation by 120◦around a line containing (0, 0, 0) and (1, 1, 1)blongs to A =0 0 11 0 00 1 0which permutes ~e1→ ~e2→ ~e3.REFLECTION AT PL ANE. To a reflection at the xy-plane belongs the matrixA =1 0 00 1 00 0 −1as can be seen by looking at the images of ~ei. The pictureto the right shows the textbook and re flec tio ns of it at two different mirrors.PROJECTION ONTO SPACE. To project a 4d-object into xyz-space, usefor example the matrix A =1 0 0 00 1 0 00 0 1 00 0 0 0. The picture shows the pro-jection of the four dimensional cube (tesseract, hypercube) with 16 edges(±1, ±1, ±1, ±1). The tesseract is the theme of the horror movie ”hypercube” .10/6/2003, LINEAR TRAFOS IN GEOMETRY Math 21b, O. KnillHW: section 2.2: 6,8,16,30,34,47*,50*LINEAR TRANSFORMATIONS DE-FORMING A BODYA CHARACTERIZATION OF LINEAR TRANSFORMATIONS: a tr ansformation T from Rnto RmwhichsatisfiesT (~0) =~0, T (~x + ~y) = T (~x) + T (~y) and T (λ~x) = λT (~x)is a linear transformation.Proof. Call ~vi= T (~ei) and define S(~x) = A~x. Then S(~ei) = T (~ei). With ~x = x1~e1+ ... + xn~en, we haveT (~x) = T (x1~e1+ ... + xn~en) = x1~v1+ ... + xn~vnas well as S(~x) = A(x1~e1+ ... + xn~en) = x1~v1+ ... + xn~vnproving T(~x) = S(~x) = A~x.SHEAR:A =1 01 1A =1 10 1In general, shears are transformation in the plane with the property that there is a vector ~w such that T ( ~w) = ~wand T (~x) − ~x is a multiple of ~w for all ~x. If ~u is orthogonal to ~w, then T (~x) = ~x + (~u · ~x) ~w.SCALING:A =2 00 2A =1/2 00 1/2One can also look at transformations which scale x differently then y and where A is a diagonal matrix.REFLECTION:A =cos(2α) sin(2α)sin(2α) − cos(2α)A =1 00 −1Any reflection at a line has the fo rm of the matrix to the left. A r e flec tio n a t a line containing a unit vector ~uis T (~x) = 2(~x · ~u)~u − ~x with matrix A =2u21− 1 2u1u22u1u22u22− 1PROJECTION:A =1 00 0A =0 00 1A projection onto a line containing unit vector ~u is T (~x) = (~x · ~u)~u with matrix A =u1u1u2u1u1u2u2u2ROTATION:A =−1 00 −1A =cos(α) − sin(α)sin(α) cos(α)Any rotation has the form of the matrix to the right.ROTATION-DILATION:A =2 −33 2A =a −bb aA rotation dilation is a composition of a rotation by angle arctan(y/x) and a dilation by a factorpx2+ y2. Ifz = x + iy and w = a + ib and T (x, y) = (X , Y ), then X + iY = z w. So a rotation dilation is tied to the processof the multiplication with a complex number.BOOST:A =cosh(α) sinh(α)sinh(α) cosh(α)The boost is a basic Lorentz transformationin s pecial 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 transformatio n 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, c t) = ((x + vt)/γ, t + (v/c2)/γx), with γ =p1 − v2/c2.ROTATION IN SPACE. Rotations in space are defined by an axes of rotationand an angle. A rotation by 120◦around a line containing (0, 0, 0) and (1, 1, 1)blongs to A =0 0 11 0 00 1 0which permutes ~e1→ ~e2→ ~e3.REFLECTION AT PL ANE. To a reflection at the xy-plane belongs the matrixA =1 0 00 1 00 0 −1as can be seen by looking at the images of ~ei. The pictureto the right shows the textbook and re flec tio ns of it at two different mirrors.PROJECTION ONTO SPACE. To project a 4d-object into xyz-space, usefor example the matrix A =1 0 0 00 1 0 00 0 1 00 0 0 0. The picture shows the pro-jection of the four dimensional cube (tesseract, hypercube) with 16 edges(±1, ±1, ±1, ±1). The tesseract is the theme of the horror movie ”hypercube”
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