LINEAR TRANSFORMATIONS Math 21b, O. KnillHOMEWORK. 2.1: 6,14,24-30,42,50,34*,44*TRANSFORMATIONS. A transformation T from a set X to a set Y is a rule, which a ssigns to every elementin X an element y = T (x) in Y . One calls X the domain and Y the codomain. A tra nsformation is alsocalled a map.LINEAR TRANSFORMATION. A map T from Rnto Rmis called a linear transformation if there is am × n matrix A such that T (~x) = A~x.EXAMPLES.• To the linear tra ns formation T (x, y) = (3x+4y, x+5y) belongs the matrix3 41 5. This transformationmaps the plane onto itself.• T (x) = −3x is a linear transformation from the real line onto itself. The matrix is A = [−3].• To T (~x) = ~y ·~x from R3to R belongs the matrix A = ~y =y1y2y3 . This 1 × 3 matr ix is also ca lleda row vector. If the codomain is the real axes, one calls the map also a function. function defined onspace.• T (x) = x~y from R to R3. A = ~y =y1y2y3is a 3 × 1 matrix which is also called a column vector. Themap defines a line in space.• T (x, y, z) = (x, y) from R3to R2, A is the 2 × 3 matrix A =1 00 10 0. The map projects space onto aplane.• To the map T (x, y) = (x + y, x − y, 2x − 3y) belongs the 3 × 2 matrix A =1 1 21 −1 −3. The imageof the map is a plane in three dimensional space.• If T (~x) = ~x, then T is called the identity transformation.PROPERTIES OF LINEAR TRANSFORMATIO NS.T (~0) =~0 T (~x + ~y) = T (~x) + T (~y) T (λ~x) = λT (~x)In 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.ON LINEAR TRANSFORMATIONS. Linea r transformations generalize the scaling transformation x 7→ ax inone dimensions.They are important in• geometry (i.e. rotations, dilations, projections or r e flections)• art (i.e. perspective, coordinate transformatio ns),• CAD 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 trans-form),• coding (many c odes are linear codes),• probability (i.e. Markov processes).• linear eq uations (inversion is solving the equation)LINEAR TRANSFORMATION OR NOT? (The square to the right is the image of the square to the left):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.In order to find the matrix of a linear transformation, look at theimage of the standard vectors and use those to build the columnsof the matrix.QUIZ.a) Find the matrix belonging to the linear transformation, which rota tes a cube around the diagonal (1, 1, 1)by 120 degrees (2π/3).b) Find the linear transformation, which reflects a vector at the line containing the vector (1, 1, 1).INVERSE OF A TRANSFORMATION. If S is a second transformation such that S(T ~x) = ~x, for every ~x, thenS is called the inverse of T . We will discus s this more later.SOLVING A LINEAR SYSTEM OF EQUATIONS. A~x =~b means to invert the linear transformation ~x 7→ A~x.If the linear system has exactly one solution, then an inverse exists. We will write ~x = A−1~b and see that theinverse of a linear transformation is again a linear transformation.THE BRETSCHER CODE. Otto Bretschers book contains as a motivation a”code”, where the encryption happens with the linear map T (x, y) = (x + 3y, 2x +5y). The map has the inverse T−1(x, y) = (−5x + 3y, 2x − y).Cryptologists use often the following approach to crack a encryption. If one knows the input and output ofsome data, one often can decode the key. Assume we know, the enemy uses a Bretscher code and we know thatT (1, 1) = (3, 5) and T (2, 1) = (7, 5). How do we get the code? The problem is to find the matrix A =a bc d.2x2 MATRIX. It is useful to decode the Bretscher code in general. If ax + by = X and cx + dy = Y , thenx = (dX − bY )/(ad − bc), y = (cX − aY )/(ad − bc). This is a linear transformation with matrix A =a bc dand the corresponding matrix is A−1=d −b−c a/(ad − bc).”Switch diagonally, negate the wings and scale with a
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