Some linear transformations on R 2 Math 3 Linear Algebra D Joce, Fall 23 Let s look at some some linear transformations on the plane R 2. We ll look at several kinds of operators on R 2 including reflections, rotations, scalings, and others. We ll illustrate these transformations b appling them to the leaf shown in figure. In each of the figures the -ais is the red line and the -ais is the blue line. Figure : Basic leaf Figure 2: Reflected across -ais Eample (A reflection). Consider the 2 2 matri A =. Take a generic point [ = (, ) in the plane, and write it as the column vector =. Then the matri product A is A = [ = Thus, the matri A transforms the point (, ) to the point T (, ) = (, ). You ll recognize this right awa as a reflection across the -ais. The -ais is special to this operator as it s the set of fied points. In other words, it s the -eigenspace. The -ais is also special as ever point (, ) is sent to its negation (, ). That means the -ais is an eigenspace with eigenvalue, that is, it s the - eigenspace. There aren t an other eigenspaces. A reflection alwas has these two eigenspaces a -eigenspace and a -eigenspace. Ever 2 2 matri describes some kind of geometric transformation of the plane. But since the origin (, ) is alwas sent to itself, not ever geometric transformation can be described b a matri in this wa. Eample 2 (A rotation). The matri A = the vector = to the vector = [ determines the transformation that sends. It determines the linear operator T (, ) =
(, ). In particular, the two basis vectors e = and e 2 = are sent to the vectors e 2 = and e 2 = respectivel. Note that these are the first and second columns of A. You ll recognize this transformation as a rotation around the origin b 9. (As alwas, the convention is to alwas take counterclockwise rotations to be b a positive number of degrees, but clockwise ones b a negative number of degrees.) There are no eigenvalues for this rotation. There are no nontrivial eigenvectors at all. (We don t consider since it s alwas sent to itself.) If, however, ou use comple scalars, there is one. The vector (, i) is sent to (i, ), so it s an i-eigenvector, and (, i) is sent to ( i, ), so it s a i-eigenvector. We ll look at comple eigenvalues later when we stud linear operators in detail. Figure 3: Rotated 9 Figure 4: Rotated Figure 5: Reflected across the line = Eample 3 (Other rotations). [ Rotations b other angles θ can be described with the help cos θ sin θ of trig functions. The matri describes a rotation of the plane b an angle sin θ cos θ of θ. For eample, the[ matri that describes a[ rotation of the plane around the origin of cos sin counterclockwise is.9848.736 sin cos = since sin.736.9848 =.736 and cos =.9848. Note that for an rotation, an circle centered at the origin is invariant. A set is said to be invariant under a transformation if it is sent to itself. The individual points in the set ma be moved, but the must sta within the set. All rotations and reflections preserve distance. That means that the distance between an two vectors v and w is the same as the distance between their images Av and Aw. Such transformations are called rigid transformations or isometries. Eample 4 (Other reflections). We ve alread seen that the matri describes a reflection across the -ais. Likewise, describes a reflection across the -ais. There s a 2 2 matri for reflection across an line through the origin. What matri describes a reflection across the line =? What are the eigenspaces for this reflection? 2
Eample 5 (Epansions). Not all linear[ operators preserve distance. For instance, [ contractions and epansions don t. The matri sends a vector to the vector Thus, 2 2 2 2 ever point is sent twice as far awa from the origin. That s an epansion b a factor of 2. Note that ever vector is a 2-eigenvector. In other words, all of R 2 is the 2-eigenspace. Figure 6: Epansion Figure 7: Contraction Figure 8: Half turn A scalar matri is a multiple of the identit matri like this one was. If λ is a scalar, then λ λi = λ = λ is a scalar matri with scalar λ. Ever scalar matri where the scalar is greater than describes an epansion. Eample 6 (Contractions). [ When the scalar is between and, then the matri describes a.5 contraction. For instance moves vectors toward the origin half as far awa as where.5 the started. Eample 7 (Point inversion a.k.a half turn). The particular scalar matri sends a point to the other side of the origin, but the same distance awa from the origin. That s the same as a 8 rotation. This operation is called a half turn or point inversion. Other scalar matrices with negative scalars describe transformations that can be thought of as compositions of point inversions and either epansions or contractions. Sometimes the terms dilation and dilatation is used for an of these transformations determined b scalar matrices. 3
Eample 8 (Projections). Consider the 2 2 matri. It sends the vector to the vector. This particular transformation is a projection from 2-space onto the -ais that forgets the -coordinate. Projections have -eigenspaces and -eigenspaces. What are the for this projection? This projection will map our leaf to the -ais as shown in figure 9. Eample 9 (Shear transformations). The matri describes a shear transformation that fies the -ais, moves points in the upper half-plane to the right, but moves points in the lower half-plane to the left. In general, a shear transformation has a line of fied points, its -eigenspace, but no other eigenspace. Shears are deficient in that wa. Note that an line parallel to the line of fied points is invariant under the shear transformation. Figure 9: Projection to -ais Figure : A shear transformation Eample [ (Stretch and squeeze). Another interesting transformation is described b the 2 2 matri which sends the vector to the vector. The plane is transformed.5.5 b stretching horizontall b a factor of 2 at the same time as it s squeezed verticall. (What are its eigenvectors and eigenvalues?) For this transformation, each hperbola = c is invariant, where c is an constant. These last two eamples are plane transformations that preserve areas of figures, but don t preserve distance. If ou randoml choose a 2 2 matri, it probabl describes a linear transformation that doesn t preserve distance and doesn t preserve area. Let s look at one more eample before leaving dimension 2. Eample (Rotar contraction). [ If ou compose a rotation and a contraction, ou get cos θ sin θ a rotar contraction. The matri describes a rotation about the origin b [ sin θ cos θ.5 an angle of θ, while the matri contracts the plane toward the origin b.5, so the 2 4
Figure : Stretch and squeeze Figure 2: A rotar contraction.5 cos θ.5 sin θ product of the matrices describes the composition of contracting and.5 sin θ.5 cos θ rotating. (In this case, the order of composing them doesn t matter since the two operations commute, but usuall order makes a difference.) The result is a kind of spiral transformation. If ou take an point and repeatedl appl this transformation, the orbit that is produced is a spiral toward the origin. Logarithmic spirals whose equations in polar coordinates r = c2 2θ/π are invariant subsets of this rotar contraction, where c is an constant. Transformations of R 3. A 3 3 matri describes a transformation of space, that is, a 3-D operator. There are man kinds of such transformations, some isometries, some not. Isometries include () reflections across planes that pass through the origin, (2) rotations around lines that pass through the origin, and (3) rotar reflections. Rotar reflections are compositions of a reflection across a plane and rotation around an ais perpendicular to that plane. Most of the linear transformations on R 3 aren t isometries. The include projections, epansions and contractions, shears, rotar epansions and contractions, and man others. Math 3 Home Page at http://math.clarku.edu/~djoce/ma3/ 5