Math 19b: Linear Algebra with Probability Oliver Knill, Spring 2011Lecture 8: Examples of linear transformationsWhile the space of linear transformations is large, there are few types of transformations whichare typical. We look here at dilations, shears, rotations, reflections and projections.Shear transformations1A ="1 01 1#A ="1 10 1#In general, shears are t r ansformation in the plane with the property that there is a vector ~w suchthat T (~w) = ~w and T (~x) −~x is a multiple of ~w for all ~x. Shear transformations are invertible,and are important in general because they are examples which can not be diagonalized.Scaling transformations2A ="2 00 2#A ="1/2 00 1/2#One can also look at transformations which scale x differently then y and where A is a diagonalmatrix. Scaling transformations can also be written as A = λI2where I2is the identity matrix.They are also called dilations.Reflection3A ="cos(2α) sin(2α)sin(2α) −cos(2α)#A ="1 00 −1#Any reflection at a line has the form of the matrix to the left. A reflection at a line containinga unit vector ~u is T (~x) = 2(~x · ~u)~u − ~x with matrix A ="2u21− 1 2u1u22u1u22u22− 1#Reflections have the property that they are t heir own inverse. If we combine a reflection witha dilation, we get a reflection-dilation.Projection4A ="1 00 0#A ="0 00 1#A projection onto a line containing unit vector ~u is T (~x) = (~x · ~u)~u with matrix A ="u1u1u2u1u1u2u2u2#.Projections are also important in statistics. Projections are not invertible except if we projectonto the entire space. Projections also have the property that P2= P . If we do it twice, itis the same transformation. If we combine a projection with a dilation, we get a rotationdilation.Rotation5A ="−1 00 −1#A ="cos(α) −sin(α)sin(α) cos(α)#Any rotation has the form of the matrix to the right.Rotations are examples of orthogonal tra nsfor mations. If we combine a rotation with a dilation,we get a rotation-dilation.Rotation-Dilation6A ="2 −33 2#A ="a −bb a#A rotation dilation is a composition of a rotation by angle arctan(y/x) and a dilation by afactor√x2+ y2.If z = x + iy and w = a + ib and T (x, y) = (X, Y ), then X + iY = zw. So a rotation dilationis tied to the process of the multiplication with a complex number.Rotations in space7Rotations in space are determined by an axis of rotation and anangle. A rotation by 120◦around a line containing (0, 0, 0) and(1, 1, 1) belongs to A =0 0 11 0 00 1 0which permutes ~e1→ ~e2→ ~e3.Reflection at xy-plane8To a reflection at the xy-plane belongs the matrix A =1 0 00 1 00 0 −1as can be seen by looking at the images of ~ei. Thepicture to the right shows the linear algebra textbook reflected attwo different mirrors.Projection into space9To project a 4d-o bject into the three dimensional xyz-space, usefor example the matrix A =1 0 0 00 1 0 00 0 1 00 0 0 0. The picture showsthe proj ection of the four dimensional cube (tesseract, hypercube)with 16 edges (±1, ±1, ±1, ±1). The tesseract is the theme of thehorror movie ”hypercube”.Homework due February 16, 20111 What transformation in space do you get if you reflect first at the xy-plane, then rotatearound t he z a xes by 90 degrees (counterclockwise when watching in the direction of thez-axes), and finally reflect at the x axes?2 a) One of the following matrices can be composed with a dilation to b ecome an orthogonalprojection onto a line. Which one?A =3 1 1 11 3 1 11 1 3 11 1 1 3B =3 1 0 01 3 0 00 0 3 10 0 1 3C =1 1 1 11 1 −1 −11 −1 1 −11 −1 −1 −1D =1 1 1 11 1 1 11 1 1 11 1 1 1E =1 1 0 01 1 0 00 0 1 10 0 1 1F =1 −1 0 01 1 0 00 0 1 −10 0 1 1b) The smiley face visible to the right is transformed with various lineartransformations represented by matrices A − F . Find out which matrixdoes which transformation:A="1 −11 1#, B="1 20 1#, C="1 00 −1#,D="1 −10 −1#, E="−1 00 1#, F="0 1−1 0#/2A-F image A-F image A-F image3 This is homework 28 in Bretscher 2.2: Each of the linear transformations in parts (a) through(e) corresponds to one and only one of the matrices A) through J). Match them up.a) Scaling b) Shear c) Rotation d) Orthogonal Projection e) ReflectionA ="0 00 1#B ="2 11 0#C ="−0.6 0.80.8 −0.6#D ="7 00 7#E ="1 0−3 1#F ="0.6 0.80.8 −0.6#G ="0.6 0.60.8 0.8#H ="2 −11 2#I ="0 01 0#J ="0.8 −0.60.6