2/14/2002, LINEAR TRANSFORMATIONS (II) Math 21b, O. KnillHOMEWORK: 2.2: 4,8,10,32,47*,50*, 2.3: 10,20,26*,30,40,42*INVERSE OF LINEAR TRANSFORMATION. If A is a n × n matrix and T : ~x 7→ Ax has an inverse S, thenS is linear and the A−1, the matrix belonging to S is called the inverse of A.FINDING THE INVERSE. Let 1nbe the n × n matrix with 1 in the diagonal and zero elsewhere. Start with[A|1n] and make Gaussian elimination. Thenrref([A|1n]) =1n|A−1 EXAMPLE. Find the inverse of A =2 61 4with Gaussian elimination. Do-ing the Gauss-Jordan elimination to theright gives A−1=2 −3−1/2 1.2 6 | 1 01 4 | 0 11 3 | 1/2 01 4 | 0 11 3 | 1/2 00 1 | −1/2 11 0 | 2 −30 1 | −1/2 1SHEAR:A =1 0−1 1A−1=1 01 1SCALING:A =2 00 3A−1=1/2 00 1/3REFLECTION:A =cos(2α) sin(2α)sin(2α) −cos(2α)A−1=cos(2α) sin(2α)sin(2α) −cos(2α)ROTATION:A =cos(α) sin(α)−sin(α) cos(−α)A−1=cos(α) −sin(α)sin(α) cos(α)ROTATION-DILATION:A =a −bb aA−1=a/r2b/r2−b/r2a/r2, r2= a2+ b2BOOST:A =cosh(α) sinh(α)sinh(α) cosh(α)A−1=cosh(α) −sinh(α)−sinh(α) cosh(α)NONINVERTIBLE EXAMPLE. The projection ~x 7→ A~x =1 00 0is a non-invertible transformation.MORE ON SHEARS. The shears T (x, y) = (x + ay, y) or T (x, y) = (x, y + ax) in R2can be generalized. Ashear is a linear transformation which fixes some line L through the origin and which has the property thatT (~x) −~x is parallel to L for all ~x.WHERE DO THEY APPEAR? Optics (see next week). Galileo transformation (x, t) 7→ (x + tv, t).PROBLEM. T (x, y) = (3x/2 + y/2, y/2 − x/2) is a shear along a line L. Find L.SOLUTION. Solve the system T (x, y) = (x, y). You find that the vector (1, −1) is preserved.MORE ON PROJECTIONS. A linear map T with the property that T(T(x)) = T (x) is a projection. Examples:T (~x) = (~y · ~x)~y is a projection onto a line spanned by a unit vector ~y.WHERE DO PROJECTIONS APPEAR? CAD: describe 3D objects using projections. A photo of an image isa projection. Compression algorithms like JPG or MPG or MP3 use projections (cut away the high frequencies).MORE ON ROTATIONS. A linear map T which preserves the angle between two vectors and the length ofeach vector is called a rotation. Rotations form an important class of transformations and will be treated laterin more detail. In two dimensions, every rotation is of the form x 7→ A(x) with A =cos(φ) −sin(φ)sin(φ) cos(φ).EXAMPLES of rotations in three dimensions are ~x 7→ Ax, with A =cos(φ) −sin(φ) 0sin(φ) cos(φ) 00 0 1, which is arotation around the z axis. or similar around any other major axes.One can also get rotations by composing two reflections at planes. The axis of rotation is the intersection ofthe planes, the angle is twice the angle between the planes.WHERE DO ROTATIONS APPEAR? Useful for example when putting an object into a form which is bettermanageable: Example: align a cylinder along the z-axis to make some computation. A rigid body centered atsome point can only rotate.MORE ON REFLECTIONS. Reflections are linear transformations different from the identity which are equalto their own inverse. Examples: in the plane reflections at the origin: A =−1 00 1, reflections at aline A =cos(2φ) sin(2φ)sin(2φ) −cos(2φ). In higher dimensions, the reflection at a line containing a unit vector ~yis T (~x) = 2(~x · ~y)~y −~x.Examples in 3D: reflections at the origin: A =−1 0 00 −1 00 0 −1. reflections at a line (for example the zaxis): A =−1 0 00 −1 00 0 1. reflections at a plane (for example the xy-plane: A =1 0 00 1 00 0 −1.WHERE DO REFLECTIONS APPEAR? Important symmetries in physics: T (time reflection), P (reflectionat a mirror), C (change of charge) are reflections. It seems today that the composition of TCP is a fundamentalsymmetry in nature.CHARACTERIZATION OF LINEAR TRANSFORMATIONS.A linear transformation T is linear if and only if T (~x + ~y) =T (~x) + T (~y) and T (λ~x) = λT (~x) for all ~x, ~y, λ.PROOF. If ~x = x1~e1+ ... + ~en, then T (x) = x1T (~e1) + ... + xnT (~en) = x1~v1+ ... + xn~vn. This can be rewrittenas T (~x) = A~x, where A is the matrix with vectors vjas columns.EXAMPLE. Using the above criterium, show that the reflection T (~x) = 2(~x ·~y)~y −~x is a linear transformation.EXAMPLE. Using the above criterium, show that the inverse of a linear transformation is linear.DETERMINANT The determinant of a 2 × 2 matrix A =a bc dis ad − bc.EXAMPLES. In the plane, rotations, boosts, shears have determinant 1, reflections have determinant -1.THE INVERSE OF LINEAR MAPS R27→ R2.There are explicit formulas for the inverse of invertible linear transformations (see later).The formula in two dimensions can be remembered:If ad −bc 6= 0, the inverse of a linear transformation ~x 7→ Ax withA =a bc dis A−1=d −b−d a/(ad − bc).This can be checked directly.PRO MEMORIAM: ”To find the inverse, flip the diagonal, change sign of the wings and divide by the deter-minant.”COMPOSING LINEAR TRANSFORMATIONS. If T : Rn→ Rm, x 7→ Ax and S : Rm→ Rk, x 7→ Bx arelinear transformations, then their composition S ◦ T is a linear transformation. (Later: The matrix of thecomposition is the matrix product BA).EXAMPLE. A rotation dilation is a composition of a rotation by an angle α = arctan(b/a) and a scale by factorr =√a2+
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