THE INVERSE Math 21b, O. KnillINVERTIBLE TRANSFORMATIONS. A map T from Xto Y is invertible if there is for every y ∈ Y a uniquepoint x ∈ X such that T (x) = y.⇒EXAMPLES.1) T (x) = x3is invertible from X = R to X = Y .2) T (x) = x2is not invertible from X = R to X = Y .3) T (x, y) = (x2+ 3x − y, x) is invertible from X = R2to Y = R2.4) T (~x) = Ax linear and rref(A) has an empty row, then T is not invertible.5) If T (~x) = Ax is linear and rref(A) = 1n, then T is invertible.INVERSE OF LINEAR TRANSFORMATION. If A is a n × n matrix and T : ~x 7→ Ax has an inverse S, thenS is linear. The matrix A−1belonging to S = T−1is called the inverse matrix of A.First proof: check that S is linear using the characterization S(~a +~b) = S(~a) + S(~b), S(λ~a) = λS(~a) of linearity.Second proof: construct the inverse using Gauss-Jordan elimination.FINDING THE INVERSE. Let 1nbe the n × n identity matrix. Start with [A|1n] and perform Gauss-Jordanelimination. Thenrref([A|1n]) =1n|A−1 Proof. The elimination process actually solves A~x = ~eisimultaneously. This leads to solutions ~viwhich are thecolumns of the inverse matrix A−1because A−1~ei= ~vi.EXAMPLE. Find the inverse of A =2 61 4.2 6 | 1 01 4 | 0 1A | 12 1 3 | 1/2 01 4 | 0 1.... | ... 1 3 | 1/2 00 1 | −1/2 1.... | ... 1 0 | 2 −30 1 | −1/2 112| A−1 The inverse is A−1=2 −3−1/2 1.THE INVERSE OF LINEAR MAPS R27→ R2:If ad−bc 6= 0, the inverse of a linear transformation ~x 7→ Ax with A =a bc dis given by the matrix A−1=d −b−d a/(ad − bc).SHEAR:A =1 0−1 1A−1=1 01 1DIAGONAL:A =2 00 3A−1=1/2 00 1/3REFLECTION:A =cos(2α) sin(2α)sin(2α) − cos(2α)A−1= A =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.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 where the high frequencies arecut away.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).An example of a rotations in three dimensions are ~x 7→ Ax, with A =cos(φ) − sin(φ) 0sin(φ) cos(φ) 00 0 1. it is a rotationaround the z axis.MORE ON REFLECTIONS. Reflections are linear transformations different from the identity which are equalto their own inverse. Examples:2D reflections at the origin: A =−1 00 1, 2D reflections at a line A =cos(2φ) sin(2φ)sin(2φ) − cos(2φ).3D reflections at origin: A =−1 0 00 −1 00 0 −1. 3D reflections at a line A =−1 0 00 −1 00 0 1. Bythe way: in any dimensions, to a reflection at the line containing the unit vector ~u belongs the matrix [A]ij=2(uiuj) − [1n]ij, because [B]ij= uiujis the matrix belonging to the projection onto the line.The reflection at a line containng the unit vector ~u = [u1, u2, u3] is A =u21− 1 u1u2u1u3u2u1u22− 1 u2u3u3u1u3u2u23− 1.3D reflection at a plane A =1 0 00 1 00 0 −1.Reflections are important symmetries in physics: T (time reflection), P (reflection at a mirror), C (change ofcharge) are reflections. It seems today that the composition of TCP is a fundamental symmetry in
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