Properties of Transformations

Transcription

1 Properties of Transformations P. Danziger Transformations from R n R m. General Transformations A general transformation maps vectors in R n to vectors in R m. We write T : R n R m to indicate this. Example. Given T : R R, T x, y x +, y, find T 0,. T 0,,.. Given T : R R, T x, y x, y, find T, T,,. 3. Given T : R R 3, T x, y x, y, x + y, find T,. T,,,. 4. Given T : R 3 R, T x, y, z x + z, y + z, find T,,. T,, 0,. Definition Given a transformation T : R n R m. The domain is all those values x R n where T x is defined. domt {x R n T x is defined }.. The range of T is the set of values in y R m for which there is an x R n such that y T x. i.e. y R n for which there exists x R m such that y T x. rant {y R n there exists x R m such that y T x}. 3. The kernal of a transformation T is the set of x R n such that T x 0 and is denoted kert.

2 . Matrix Transformations We can interpret a matrix as a map. Given an m n matrix A, for each x R n define y R m by y Ax. Thus each m n matrix A is associated with a transformation T A where T A x Ax. A transformation which comes from an associated matrix is called a matrix transformation Given a matrix transformation T A, we write Notes: A [T A ]. If T is a matrix transformation associated with an m n matrix A then domt R n. If T is a matrix transformation then finding the kernal is equivalent to solving the homogeneous system Ax 0. Example 3. Let A a Find the image of the vector,,. 0 0 b Find T A,, 0. So T A,, 0,, c Find domt A. T A is a matrix transformation, so domt A R 3. 0 d Find kert A. Find x R 3 such that T x 0, i.e. Solve Ax R 3 R 3 R Only solution is the trivial solution x x x 3 0. So kert A {0}

3 e Find rant A. We must find all y R 3 so that y Ax for some x R 3. i.e. Solve Ax y. 0 y 0 y R 3 R 3 R y 3 0 y 0 y 0 0 y 3 y. Which has a unique solution for every y R 3. So rant A R 3. a Find kert A. Let A R 3 R 3 R R 3 R 3 R Solution: Let t R, x 3 t, y t, x t, or x t,,, which is a line parallel to,, through the origin. b Find rant A. kert A {x R 3 x t,,, t R 3 } 0 y 0 y R 3 R 3 R y 3 0 y 0 y R 3 R 3 R 0 y 3 y 0 y 0 y y 3 y y Which will have solution only when y + y y 3 0. rant A {x, y, z R 3 x + y z 0} So the range of T A is the plane x + y z 0 and T A maps R 3 to this plane. 3

4 c Is,, 0 in the range of T A? From the above equation the vector x, y, z is in the range of T A if and only if x+y z 0. In this case, x, y, z,, 0, we have So,, 0 rant A. If we did not have the work above we would be asking to solve Ax R 3 R 3 R R 3 R 3 R Which has no solution, so the original vector,, 0 is not in the range. d Is,, in the range of T A? Once again given the answer to part b we can see that + 0, so,, rant A. If we did not have the answer to part b we would proceed as follows. 0 0 R 3 R 3 R 0 0 R 3 R 3 R Which has solution set t, t, t, t R. So,, rant A. e Describe the transformation T A. This transformation maps R 3 to the plane x+y z 0 along lines parallel to t,,. Theorem 4 Given an m n matrix A, the range of T A is a subspace of R m and the kernel of T A is a subspace of R n. 4

5 .3 Linear Transformations Definition 5 Given a transformation T : R n R m, T is called a Linear Transformation if for every u, v R n and every scalar c the following two properties hold:. T u + v T u + T v.. T cu ct u. Theorem 6 A Transformation T is a linear transformation if and only if for every u, v R n and every pair of scalars c and d T cu + dv ct u + dt v. Proof: If T is a linear transformation then the result follows from properties and. T cu + dv T cu + T cv ct u + dt v. Suppose that T satisfies T cu + dv ct u + dt v. for every u, v R n and every pair of scalars c and d In particular when c d we get T u + v T u + T v. Which is property. When v 0 we get property, T cu ct u. Theorem 7 A transformation is linear if and only if it is a matrix transformation Proof: Suppose T T A for some matrix A, we will show that T is linear. Let u, v R n and c be a scalar. Consider T A cu + dv Acu + dv Acu + Adv ct A u + dt A v We will show this in due course. Theorem 8 If a transformation is linear then T 0 0. Proof:We must have T u T u + 0 T 0 + T u. Which implies T 0 0. Notes This Theorem can only be used to show that a transformation is not Linear, by showing that T 0 0. It is possible to have a transformation for which T 0 0, but which is not linear. Thus, it is not possible to use this theorem to show that a transformation is linear, only that it is not linear. To show that a transformation is linear we must show that the rules and hold, or that T cu + dv ct u + dt v. Example 9. Show that T : R R, T x, y x +, y is not linear. T 0, 0, 0 0, 0. So this transformation is not linear. 5

6 . Show that T : R R, T x, y x, y is not linear. T 0, 0 0, 0 so Theorem 8 does not apply. Now consider T, + T,, +,,. But T, +, T 0, 0,. So rule does not apply to this transformation. 3. Show that the transformation T : R 4 R, given by T x, x, x 3, x 4 x + x, x 3 + x 4 is linear. Let u u, u, u 3, u 4 and v v, v, v 3, v 4 be vectors in R 4 and c and d be scalars. Consider T cu + dv T cu + dv, cu + dv, cu 3 + dv 3, cu 4 + dv 4 cu + dv + cu + dv, cu 3 + dv 3 + cu 4 + dv 4 cu + u + dv + v, cu 3 + u 4 + dv 3 + v 4 cu + u, u 3 + u 4 + dv + v, v 3 + v 4 ct u, u, u 3, u 4 + dt v, v, v 3, v 4 ct u + dt v Theorem 0 Translations are not linear. Translations are maps of the form T x x + a, for some fixed non zero vector a R n. Consider T a a 0. Theorem Linear transformations map subspaces to subspaces..4 Finding the Matrix Associated with a Linear Transformation Given any linear transformation T : R n R m The matrix A such that T T A is the matrix whose columns are the vectors T acting on each of the elementary vectors. w w... w n So A w T e, w T e,..., w n T e n, T e T e... T e n. Given a vector u u, u,..., u n R n, we can write u u e + u e u n e n Now Au w w... w n u u. u n u w + u w u n w n 6

7 To see that T u Au consider T u T u e + u e u n e n T u e + T u e T u n e n u T e + u T e u n T e n u w + u w u n w n Au Example Find a matrix for the transformation T : R 4 R, given by T x, x, x 3, x 4 x + x, x 3 + x 4. For example T e T, 0, 0, 0 T e T 0,, 0, T e 3 T 0, 0,, 0 T e 4 T 0, 0, 0,. 0 0 So A 0 0 T,, 3, Differentiation as a linear Operator In calculus we have the following Theorem: Theorem 3 Given any two differentiable functions f and g and a constant c R, df + g dx dcf dx df dx + dg dx c df dx Which says that differentiation is a linear operator. For somplicity we restrict ourselves to polynomials, but note that by Taylors Theorem, any differentiable function can be approximated by a polynomial of sufficiently high degree. Let P n {Polynomials of degree at most n} {a 0 + a x + a x a n x n a 0, a,..., a n R} Now, any polynomial a 0 +a x+a x +...+a n x n is uniquely represented by the vector a 0, a,..., a n R n+, so there is a correspondence between the set of polynomials of degree at most n, P n and R n+, by a 0 + a x + a x a n x n P n a 0, a,..., a n R n+. 7

8 Given a polynomial px a 0 + a x + a x a n x n, dp dx a + a x na n x n P n. We can consider the corresponding map D : R n+ R n given by Da 0, a, a..., a n a, a,..., na n. Now to find the matrix associated with this map D we consider the effect of D on the standard basis vectors e i. De 0 De i ie i i > So the matrix associated with the differentiation operator D, is the n n matrix [D] n A Library of Transformations We want to build up a library of simple transformations, just as we have a library of simple functions. There are two special transformations: The Identity Transformation The Zero Transformation T I : R n R n, T I x x for every x R n. T 0 : R n R n, T 0 x 0 for every x R n. Any linear matrix transformation can be made up by taking compositions of the following basic types of transformation:. Dilations. Rotations 3. Reflections 4. Projections 5. Shears We have already seen that composition of matrix maps is equivalent to matrix multiplication. that is T A T B T AB or equivalently T A T B x T AB x. 8

9 . Dilations A dilation makes every vector bigger or smaller by a constant factor k. Thus a dilation of a vector x R n looks like D k x kx kix. Thus the associated matrix [D k ] ki. Example 4. Find a matrix for the transformation in R which shrinks every vector to half its original size. I 0. Find a dilation by a factor of in R 3 I 3. Standard Transformations in R We now consider some basic standard transformations in R.. Rotations in R Consider a counter-clockwise rotation about the origin in R. R θ i sin θ θ cos θ i R θ j sin θ θ j cos θ So R θ i cos θ, sin θ and R θ j sin θ, cos θ, and so cos θ sin θ [R θ ] sin θ cos θ 9

10 . Reflections in R x, y x, y x, y x, y x, y x, y y x F x x, y x, y F y x, y x, y F, x, y x, y Reflection about x axis Reflection about y axis Reflection about the line y x Reflection about the x axis. F x, 0, 0 and F x 0, 0,, so Reflection about the y axis. F x x, y x, y [F x x, y] F y, 0, 0 and F y 0, 0,, so Reflection about the line y x. 0 0 F y x, y x, y [F y x, y] 0 0 F, x, y y, x F,, 0 0, and F, 0,, 0, so [ F, x, y ] 0 0 Suppose that we wish to find the matrix for the reflection about an arbitrary line parallel to a vector u, which makes an angle of θ with the x axis. Now R θ u maps u parallel to the x axis and so in this transformed coordinate system reflection about u is reflection about the x axis. Finally we transform back using R θ. 0

11 Thus F u R θ F x R θ, or Rx, y R θ F x R θ x, y But composition of maps is matrix multiplication, so [R u ] [R θ ][F x ][R θ ] cos θ sin θ 0 cos θ sin θ sin θ cos θ 0 sin θ cos θ cos θ sin θ 0 cos θ sin θ sin θ cos θ 0 sin θ cos θ cos θ sin θ cos θ sin θ sin θ cos θ sin θ cos θ cos θ sin θ cos θ sin θ cos θ sin θ cos θ sin θ cosθ sinθ sinθ cosθ Where we have used the double angle formulas: cos θ cos θ sin θ, sin θ cos θ sin θ Example 5 Find the matrix for a reflection about the line tv, where v 3,. v makes an angle of θ arctan 3 with the x axis. Further, since the second coordinate is negative, v lies in the fourth quadrant. So θ π π. 6 6 cosθ sinθ cos π sin π F v 6 6 sinθ cosθ sin π 6 cos π 6 cos π sin π 3 3 sin π 3 cos π Projections in R x, y 0, y x, y x, 0 P x x, y x, 0 P y x, y 0, y Projection onto the x axis Projection onto the y axis

12 Projection onto the x axis P x, 0, 0 and P x 0, 0, 0, so Projection onto the y axis P x x, y x, 0 [P x x, y] P x, 0 0, 0 and P x 0, 0,, so P y x, y 0, y [P x x, y] Suppose that we wish to find the matrix for the projection onto an arbitrary line parallel to a vector u, which makes an angle of θ with the x axis. As above R θ u maps u parallel to the x axis and so in this transformed coordinate system reflection about u is projection onto the x axis. Finally we transform back using R θ. Thus P u R θ P x R θ, or Rx, y R θ P x R θ x, y But composition of maps is matrix multiplication, so [P u ] [R θ ][P x ][R θ ] cos θ sin θ 0 cos θ sin θ sin θ cos θ 0 0 sin θ cos θ cos θ sin θ 0 cos θ sin θ sin θ cos θ 0 0 sin θ cos θ cos θ sin θ cos θ sin θ sin θ cos θ 0 0 cos θ cos θ sin θ cos θ sin θ sin θ Example 6 Find the matrix for a projection onto the line tv, where v 3,. As above v makes an angle of θ arctan 3 with the x axis. Further, since the second coordinate is negative, v lies in the fourth quadrant. So θ π π. 6 6 cos P v θ cos θ sin θ cos π cos π 6 6 sin π 6 cos θ sin θ sin θ cos π 6 sin π sin π

13 4. Shears in R A shear has exactly one off diagonal entry non-zero, the diagonal entries are all ones. Thus shears in R have one of two forms, we consider the action on the unit square: k,, 0 0, k, k > 0 k > 0 k 0 0 k T, 0, 0, T 0, k, T, 0, k, T 0, 0, Shear in x direction with factor k.3 Standard Transformations in R 3 Shear in y direction with factor k We now consider some basic standard transformations in R 3. A general description of transformations in R 3 is complicated, we consider only some standard ones.. Rotations in R 3 For the sense of a rotation in R 3 we use the right hand rule: If the rotation is in the direction of the fingers of your right hand, then the axis is along your thumb. Similarly, if the axis is along the thumb of your right hand, then the rotation will be in the direction of the fingers on your right hand. Rotation about the x axis in R 3 In this case the yz plane is rotated by theta, and the x coordinate is unchanged. We can thus use the rotation in R for the yz part. [R x,θ ] cos θ sin θ 0 sin θ cos θ 3

14 Rotation about the y axis in R 3 In this case the sense of the rotation is reversed, so it corresponds to a rotation in the xz plane of θ. cos θ 0 sin θ [R y,θ ] 0 0 sin θ 0 cos θ Rotation about the z axis This case is equivalent to a rotation in the xy plane by θ. [R z,θ ] cos θ sin θ 0 sin θ cos θ For completeness we give the matrix for a generalized rotation about an axis given by a vector u a, b, c by an angle θ: a cos θ + cos θ ab cos θ c sin θ ac cos θ + b sin θ [R z,θ ] ab cos θ + c sin θ b cos θ + cos θ bc cos θ a sin θ ac cos θ b sin θ bc cos θ + a sin θ c cos θ + cos θ Theorem 7 If T, T,..., T k is a succession of rotations about axes through the origin in R 3, then the resulting mapping can be obtained by a single rotation about some suitable axis through the origin.. Reflections in R 3 Reflection about xy plane F xy x, y, z x, y, z F xy, 0, 0, 0, 0, F xy 0,, 0 0,, 0, F xy 0, 0, 0, 0,. [F xy ] Reflection about xz plane F xz x, y, z x, y, z F xz, 0, 0, 0, 0, F xz 0,, 0 0,, 0, F xz 0, 0, 0, 0,. [F xz ]

15 Reflection about yz plane F yz x, y, z x, y, z F yz, 0, 0, 0, 0, F yz 0,, 0 0,, 0, F yz 0, 0, 0, 0,. 3. Projections in R 3 [F yz ] Projection onto the xy plane P xy x, y, z x, y, 0 P xy, 0, 0, 0, 0, P xy 0,, 0 0,, 0, P xy 0, 0, 0, 0, 0. [P xy ] Projection onto the xz plane P xz x, y, z x, 0, z P xz, 0, 0, 0, 0, P xz 0,, 0 0, 0, 0, P xz 0, 0, 0, 0,. [P xz ] Projection onto the yz plane P yz x, y, z 0, y, z P yz 0, 0, 0, 0, 0, P yz 0,, 0 0,, 0, P yz 0, 0, 0, 0,. [P yz ] 3 - and Onto Transformations Definition 8 Given a transformation T : R n R m T is called one to one - or injective if For every u, v R n, T u T v u v. i.e. T maps distinct vectors to distinct vectors.. T is called onto or surjective if for all y R m, there exists x R n such that y T x. 3. If T is both one to one and onto injective and surjective it is called a a bijection. 5

16 Notes Given a matrix transformation, T A : R n R m : T A is one to one if and only if for every b R m there is a unique x R n such that Ax b. A is - if and only if Ax b has unique solution for every b R m. T A is one to one if and only if kert A {0}. A is - if and only if Ax 0 has only the trivial solution x 0. T A is onto if and only if rant A R m. A is onto if and only if Ax b is consistent for every b in R m. Recall that the rank of a matrix A, ra, is the number of leading ones in the REF of A. Theorem 9 Given an m n matrix A, the corresponding matrix transformation T A : R n R m T A is - if and only if ra number of columns of A. T A is onto if and only if ra number of rows of A. Where ra is the rank of A Example 0. Is T A - or onto, where A ? R R R R 3 R 3 3R R 3 R 3 R Infinite solutions, so T A is not -. rant A R 3, so no onto. Alternately ra < 3. Thus A is neither - nor onto.. Is T A - or onto, where A? So ra. Thus T A is not -. T A, is onto. R R 3R 6

17 3. Is T A - or onto, where A 0? So ra. Thus T A is -, but not onto. 0 R 3 R 3 R 0 R 3 R 3 R Inverse Transformations Given a transformation T : R n R m, the inverse transformation, denoted T, is the transformation which undoes the action of T. So for any x domt, T T x x. Note that not every transformation has an inverase. If the inverse exists, the transformation is called invertable. If T is a matrix transformation, T A, then T A is invertable if and only if A is a square matrix and A is invertible. Notes If T A is invertible then it is both - and onto. If A is a square matrix and T A is - then A is both onto and invertible. If T A is invertible then T A T A. Example Find the inverse transformation for dilation by a factor k in R 3. T x, y, z kx, ky, kz, for some fixed k R. This corresponds to the matrix transformation T A, where k 0 0 [T A ] A 0 k k 0 0 A k 0 0. k 0 0 k Which gives the inverse transformation 7

18 5 Orthogonal Operators Recall that a set of vectors {v i { i n} is orthogonal if v i v j 0 whenever i j and 0 i j orthonormal if v i v j δ ij, where δ ij is the Kronecker delta. Note that [δ i j ij ] I. So an orthonormal set is an orthogonal set of unit vectors. Definition A linear transformation T : R n R m is orthogonal if T x x for every x R n. So orthogonal transformations preserve distances. Theorem 3 A linear transformation T : R n R m is orthogonal i.e. T x x for every x R n if and only if T x T y x y for all x, y R n. Now, since the angle θ between two vectors u and v is given by θ arccos u v. If T is an u v orthogonal transformation, the angle between the images of u and v, T u and T v is given by T u T v u v arccos arccos θ T u T v u v So orthogonal transformations preserve angles. Definition 4 A square matrix A is called orthogonal if and only if A A T We want to be sure that these two definitions are compatible. Theorem 5 Given a square matrix A the following are equivalent:. AA T I. A is an orthogonal matrix.. Ax x for all x R n. T A is an orthogonal transformation. 3. Ax Ay x y. 4. The column vectors of are orthonormal. The column vectors of A form an orthonormal basis of R n. Proof: Suppose A is orthogonal, so AA T I, and consider 3 This is Theorem 3. Ax Ax Ax x A T Ax x Ix x x x 8

19 3 4 Suppose T A is a map with Ax Ay x y for all x R n. By Theorem 3 this means that T A preserves distances and angles. In particular an orthonormal set is mapped to an orthonormal set. Now, the columns of A are given by A T e T e... T e n. {e i i n} are an orthonormal set of vectors, ans thus so are the columns of A. 4 Suppose that A { is a matrix whose column vectors, c, c,... c n, are orthonormal, so c i c j 0 i j δ ij, where δ ij i j. Now, the rows of AT are the columns of A, so A T A [c i c j ] [δ ij ] I In particular we have the following Theorem 6 A transformation T : R n R n is orthogonal if and only if its corresponding matrix [T ] is orthogonal, i.e. [T ] [T ] T The transformations which preserve distances and angles are rotations and reflections, so orthogonal transformation are exactly these. In fact in higher dimensions we define rotations to be those transformations T for which det[t ]. Definition 7 A square matrix A represents a rotation if and only if deta. Theorem 8 If A and B are orthogonal matrices, then. AB is orthogonal.. A is orthogonal. 3. A T is orthogonal. Proof: Let A and B be orthogonal matrices, so A T A and AA T BB T I. ABAB T ABB T A T AIA T AA T I. A A T A T 3 A T A A A T T Example 9 Show that the matrix A given below is orthogonal. 9

20 A AA T I The column vectors of A are Note that c i c j δ ij. c, 3, 0 c 3, 3, c 3 3,, 3 0