Math 19b: Linear Algebra with Probability Oliver Knill, Spring 2011Lecture 7: Linear transformationsA transformation T from a set X to a set Y is a rule, which assigns to every xin X an element y = T (x) in Y . One calls X the domain and Y the codomain.A transformation is also called a map from X to Y . A map T from Rmto Rniscalled a linear transformation if t here is a n × m matrix A such that T (~x) = A~x.• To the linear transformation T (x, y) = (3x + 4y, x + 5y) belongs the matrix"3 41 5#. Thistransformation maps the two-dimensional plane onto itself.• T (x) = −33x is a linear transformation from the real line onto itself. The matrix is A =[−33].• To T (~x) = ~y · ~x from R3to R belongs the mat rix A = ~y =hy1y2y3i. This 1 × 3matrix is also called a row vector. If the codomain is the real axes, one calls the map alsoa function.• T (x) = x~y from R to R3. A = ~y =y1y2y3is a 3 × 1 matrix which is also called a columnvector. The map defines a line in space.• T (x, y, z) = (x, y) fro m R3to R2, A is the 2 × 3 matrix A =1 00 10 0. The map projectsspace onto a plane.• To the map T (x, y) = (x + y, x − y, 2x − 3y) belongs the 3 × 2 mat rix A ="1 1 21 −1 −3#.The imag e of the map is a plane in three dimensional space.• If T (~x) = ~x, then T is called the identity transformation.A transfor ma t ion T is linear if and o nly if the following properties are satisfied:T (~0) =~0 T (~x + ~y) = T (~x) + T (~y) T (λ~x) = λT (~x)In other words, linear transformations are compatible with addition, scalar multiplication and alsorespect 0. It does not matter, whether we add two vectors before the transformation or add thetransformed vectors.Linear transformations are important in• geometry (i.e. rotations, dilations, projections or r eflections)• art (i.e. p erspective, coordinate transformations),• computer adided design applications (i.e. projections),• physics (i.e. Lorentz transformations),• dynamics (linearizations of general maps are linear maps),• compression (i.e. using Fourier transform or Wavelet transfor m),• error correcting codes (many codes are linear codes),• probability theory ( i.e. Markov processes).• linear equations (inversion is solving the equation)A linear transformation T (x) = Ax with A =| | · · · |~v1~v2· · · ~vn| | · · · |has the property that thecolumn vector ~v1, ~vi, ~vnare the images of the standard vectors ~e1=1···0, ~ei=0·1·0, and~en=0···1.In order to find the matrix of a linear transformation, loo k at the image of thestandard vectors and use those to build the columns of the matrix.1Find the matrix belonging to the lineartransformation, which rota t es a cube aroundthe diagonal (1, 1, 1) by 120 degrees (2π/3).2 Find the linear transformation, which reflects a vector at the line containing the vector(1, 1, 1).If there is a linear transformation S such that S(T ~x) = ~x for every ~x, then S is calledthe inverse of T . We will discuss inverse transformations later in more detail.A~x =~b means to invert the linear transforma t ion ~x 7→ A~x. If the linear system has exactlyone solution, then an inverse exists. We will write ~x = A−1~b a nd see that the inverse of alinear transformation is again a linear transformation.3 Otto Bretscher’s book contains as a motivation a ”code”, where the encryption happenswith the linear map T (x, y) = (x + 3y, 2x + 5y). It is an variant of a Hill code. The maphas the inverse T−1(x, y) = (−5x + 3y, 2x − y).Assume we know, the other party uses a Bretscher code and can find out that T (1, 1) = (3, 5)and T (2, 1) = (7, 5). Can we reconstruct the code? The problem is to find the matrixA ="a bc d#. It is useful to decode the Hill code in general. If ax+by = X and cx+dy = Y ,then x = (dX − bY )/(ad − bc), y = (cX − aY )/(ad − bc). This is a linear transformationwith matrix A ="a bc d#and the correspo nding matrix is A−1="d −b−c a#/(ad − bc).”Switch diagonally, negate the wings and scale with a cross”.Homework due February 16, 20111This is Problem 24-40 in Bretscher: Consider the circular f ace in the accompanying figure.For each of the matrices A1, ...A6, draw a sketch showing the effect of the linear transfor-mation T (x) = Ax on this face.A1="0 −11 0#. A2="2 00 2#. A3="0 11 0#. A4="1 00 −1#. A5="1 00 2#.A6="−1 00 −1#.2 This is problem 50 in Bretscher. A goldsmith uses a platinum a lloy and a silver alloy tomake jewelry; the densities of these alloys are exactly 20 and 10 grams per cubic centimeter,respectively.a) King Hiero of Syracuse orders a crown from this goldsmith, with a total mass of 5 kilograms(or 5 ,000 grams), with the stipulation that the platinum alloy must make up at least 90%of the mass. The goldsmith delivers a beautiful piece, but the king’s friend Archimedes hasdoubts about its purity. While taking a bath, he comes up with a method to check thecomposition of the crown (famously shouting ”Eureka!” in the process, and running to theking’s palace naked). Submerging the crown in water, he finds its volume to be 370 cubiccentimeters. How much of each alloy went into this piece (by mass)? Is this goldsmith acrook?b) Find the matrix A that transforms the vector"mass of platinum alloymass of silver alloy#into the vector"totalmasstotalvolume#for any piece of jewelry this goldsmith makes.c) Is the matrix A in part (b) invertible? If so, find the its inverse. Use the result to checkyour answer in part a)3 In the first week we have seen how to compute the mean and standard deviation of data.a) Given some data (x1, x2, x3, ..., x6). Is the transformation from R6→ R which maps thedata to its mean m linear?b) Is the map which assigns to the data the standard deviation σ a linear map? c) Isthe map which assigns to the data the difference (y1, y2, ..., y6) defined by y1= x1, y2=x2− x1, ..., y6= x6− y5linear? Find its matrix. d) Is the map which assigns to the data thenormalized data ( x1− m, x2− m, ..., xn− m) given by a linear