0.1 Linear Transformations

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1 .1 Linear Transformations A function is a rule that assigns a value from a set B for each element in a set A. Notation: f : A B If the value b B is assigned to value a A, then write f(a) = b, b is called the image of a under f. A is called the domain of f and B is called the codomain. The subset of B consisting of all possible values of f as a varies in the domain is called the range of f. Definition 1 Two functions f 1 and f 2 are called equal, if their domains are equal and f 1 (a) = f 2 (a) for all a in the domain Example 1 function example description f(x) f(x) = x 2 Function from R to R f(x, y) f(x, y) = x + y Function from R 2 to R f(x, y, z) f(x, y, z) = x + y + z Function from R 3 to R f(x, y, z) f(x, y, z) = (x + y, z) Function from R 3 to R 2 Functions from R n to R m If f : R n R m, then f is called a map or a transformation. If m = n, then f is called an operator on R n. Let f 1, f 2,... f m functions from R n to R, assume f 1 (x 1, x 2,..., x n ) = w 1 f 2 (x 1, x 2,..., x n ) = w 2. f m (x 1, x 2,..., x n ) = w m then the point (w 1, w 2,..., w m ) R m is assigned to (x 1, x 2,..., x n ) R n and thus those functions define a transformation from R n to R m. Denote the transformation T : R n R m and T (x 1, x 2,..., x n ) = (w 1, w 2,..., w m ) Example 2 f 1 (x 1, x 2 ) = x 1 + x 2, f 2 (x 1, x 2 ) = x 1 x 2 define an operator T : R 2 R 2. T (x 1, x 2 ) = (x 1 + x 2, x 1 x 2 ) Linear Transformations In the special case where the functions f 1, f 2,..., f m transformation T : R n R m is called a linear transformation. are linear, the 1

2 A linear transformation is defined by equations or in matrix notation or a 11 x 1 + a 12 x a 1n x n = w 1 a 21 x 1 + a 22 x a 2n x n = w a m1 x 1 + a m2 x a mn x n = w m a 11 a a 1n a 21 a a 2n... a m1 a m2... a mn Ax = w x 1 x 2. x n = The matrix A is called the standard matrix for the linear transformation T, and T is called multiplication by A. Remark: Through this discussion we showed that a linear transformation from R n to R m correspond to matrices of size m n. One can say that to each matrix A there corresponds a linear transformation T : R n R m, and to each linear T : R n R m transformation there corresponds an m n matrix A. Example 3 Let T : R 3 R 2 defined by can be expressed in matrix form as 2x 1 + 3x 2 + ( 1)x 3 = w 1 x 1 + x 2 + ( 1)x 3 = w 2 [ x 1 x 2 x 3 The standard matrix for T is [ The image of a point (x 1, x 2, x 3 ) can be found by using the defining equations or by matrix multiplication. [ 1 [ T (1, 2, ) = 2 8 = = [ w1 w 2 w 1 w 2. w m 2

3 Notation: If T : R n R m is a multiplication by A, and if it important to emphasize the standard matrix then we shall denote the transformation by T A : R n R m. Thus T A (x) = Ax Since linear transformations can be identified with their standard matrices we will use [T as symbol for the standard matrix for T : R n R m. T (x) = [T x or [T A = A Geometry of linear Transformations A linear transformation T : R n R m transforms points in R n into new points in R m Example 4 Zero Transformation The zero transformation from T : R n R m has standard matrix, so that for all x R n T (x) = Example 5 Identity Transformation The identity transformation T I : R n R m has standard matrix I n (n n identity matrix), so that T I (x) = I n x = x for all x R n. Among the more important transformations are those that cause reflections, projections, and rotations Example 6 Reflections Consider T : R 2 R 2 with standard matrix [ 1 1 then T (x) = [ 1 1 [ x1 x = x 2 T reflects points (x 1, x 2 ) about the y-axis. What might be the standard matrix of the linear transformation reflecting point about the x-axis? R 2 R 2 3

4 Operator Equation Standard matrix Reflection about the y-axis T (x, y) = ( x, y) Reflection about the x-axis T (x, y) = (x, y) Reflection about the line y = x T (x, y) = (y, x) [ 1 1 [ 1 1 [ 1 1 R 3 R 3 Operator Equation Standard matrix Reflection about the xy-plane T (x, y, z) = (x, y, z) Reflection about the xz-plane T (x, y, z) = (x, y, z) Reflection about the yz-planes T (x, y, z) = ( x, y, z) Example 7 Projections Consider T : R 2 R 2 with standard matrix [ 1 then T (x) = [ 1 [ x1 x = It gives the orthogonal projection of point (x, y) onto the x-axis. Consider T : R 3 R 3 with standard matrix then T (x) = x = It gives the orthogonal projection of a point (x, y, z) onto the xy-plane. 4 x 1 x 2

5 Example 8 Rotation: An operator that rotates a vector in R 2 through a given angle θ is called a rotation operator in R 2. T R (x) = (x cos θ y sin θ, x sin θ + y cos θ) The standard matrix of a rotation operator in R 2 for angle θ is therefore [ cos θ sin θ [T R = sin θ cos θ Proof: Let (w 1, w 2 ) = T R (x), then (check the diagram) w 1 = r cos(θ + ϕ), w 2 = r sin(θ + ϕ) 5

6 Using trigonometric identities Also (check diagram) w 1 = r cos(θ) cos(ϕ) r sin(θ) sin(ϕ) w 2 = r sin(θ) cos(ϕ) + r cos(θ) sin(ϕ) x = r cos ϕ, substituting the later into the equations above gives y = r sin ϕ w 1 = x cos(θ) y sin(θ), w 2 = x sin(θ) + y cos(θ) therefore T R (x, y) = [ w1 w 2 [ cos θ sin θ = sin θ cos θ [ x y Example 9 The standard matrix of the rotation by π/2 is [T R = [ 1 1 therefore T R (1, 2) = [ 2 1 The standard matrix of the rotation by π/4 is [ 1/ 2 1/ 2 [T R = 1/ 2 1/ 2 therefore T R (1, 2) = [ 1/ 2 3/ 2 Rotation in R 3 Operator Equation Standard matrix Counterclockwise rotation about the positive x-axis through an angle θ Counterclockwise rotation about the positive y-axis through an angle θ Counterclockwise rotation about the positive z-axis through an angle θ T (x, y, z) = T (x, y, z) = T (x, y, z) = 6 x y cos θ z sin θ y sin θ + z cos θ x cos θ + z sin θ y x sin θ + z cos θ x cos θ y sin θ x sin θ + y cos θ z 1 cos θ sin θ sin θ cos θ cos θ sin θ 1 sin θ cos θ cos θ sin θ sin θ cos θ 1

7 Dilation and Contraction This is the operator stretching or shrinking a vector by a factor k, but keeping the direction unchanged. We call the operator a dilation if the transformed vector is at least as long as the original vector, and a contraction if the transformed vector is at most as long as the original vector. Operator Equation Standard matrix Contraction with factor k on R 2, k 1) Dilation with factor k on R 2, k 1) T (x, y) = T (x, y) = [ kx ky [ kx ky [ k k [ k k Operator Equation Standard matrix Contraction with factor k on R 3, k 1) Dilation with factor k on R 3, k 1) T (x, y, z) = T (x, y, z) = kx ky kz kx ky kz k k k k k k Composition of Linear Transformations Let T A : R n R k and T B : R k R m be linear transformations, then for each x R n one can first compute T A (x), which is a vector in R k and then one can compute T B (T A (x)), which is a vector in R m. Thus the application of first T A and then of T B is a transformation from R n to R m. It is called the composition of T B with T A. and is denoted as T B T A (read T B circle T A ). (T B T A )(x) = T B (T A (x)) = T B (Ax) = B(Ax) = BAx Therefore the standard matrix of the composition of T B with T A is BA. T B T A = T BA Remark: This equation points out an important interpretation of the matrix product. Composition of two linear transformations is equivalent to the multiplication of two matrices. Example 1 In general: Composition is not commutative. T 1 :reflection about y = x, and T 2 orthogonal projection onto y 7

8 One can easily generalize the concept to the composition of more than two transformations. 8

9 .2 Properties of linear Transformations One-to-One Linear Transformations Transformations that transform different vectors into different images, that is If x y therefore T (x) T (y), are of special interest. One such example is the rotation by an angle θ in R 2. But the orthogonal projection onto the xy plane in R 3 does not have this property. T (x 1, x 2, x 3 ) = (x 1, x 2, ), so T (2, 1, 1) = T (2, 1, 45). Definition 2 A linear transformation T : R n R m is said to be one-to-one, if it is true that x y T (x) T (y) distinct vectors in R n are mapped into distinct vectors in R m. Conclusion: If T is one-to-one and w is a vector in the range of T, then there is exactly one vector in R n with T (x) = w. Consider transformations from R n to R n, then the standard matrices are square matrices of size n n. Theorem 1 If A is a n n matrix and T A : R n R n is the multiplication by A, then the following statements are equivalent (a) A is invertible (b) The range of T A is R n (c) T A is one-to-one. Proof: (a) (b) (b) (c) (c) (a) Application: The rotation by θ in R 2 is one-to-one. The standard matrix of this operator is [ cos θ sin θ A = sin θ cos θ 9

10 then det(a) = cos 2 θ + sin 2 θ = 1, therefore A is invertible and therefore the rotation operator is one-to-one. Show yourself using this criteria that the orthogonal projection in R 3 is NOT one-to-one. Inverse of a one-to-one Operator Definition 3 If T A : R n R n is a one-to-one operator, then T 1 = T A 1 is called the inverse operator of T A. Example 11 Let T : R 2 R 2 with Then and since and [T = [ T (x 1, x 2 ) = (x 1 x 2, 2x 1 ) for (x 1, x 2 ) R 2 [T 1 = 1 2 [ T 1 (x 1, x 2 ) = ( x 2 /2, (2x 1 + x 2 )/2). Theorem 2 Let T : R n R n be a one-to-one operator, then (a) T (x) = w T 1 (w) = x (b) For x R n it is (T T 1 )(x) = x, and (T 1 T )(x) = x Proof: (a) If T : R n R n is a one-to-one operator and T (x) = w, then the standard matrix [T is invertible and T (x) = w [T x = w [T 1 [T x = [T 1 w I n x = [T 1 w x = [T 1 w x = T 1 (w) 1

11 (b) If T : R n R n is a one-to-one operator and T (x) = w, then the standard matrix A is invertible and T T 1 = T A T A 1 = T AA 1 = T I therefore the claim holds. Theorem 3 A transformation T : R n R m is linear if and only if the following relationship holds for all vectors u, v R n and c R. 1. T (u + v) = T (u) + T (v) 2. T (cu) = ct (u) Example 12 Show that is a linear transformation. T (x 1, x 2 ) = (x 1 x 2, 2x 1 ) for (x 1, x 2 ) R 2 (a) Let u, v R 2, then T (u + v) = (u 1 + v 1 (u 2 + v 2 ), 2(u 1 + v 1 )) = (u 1 u 2, 2u 1 ) + (v 1 v 2, 2v 1 ) = T (u) + T (v) (b) T (cu) = (cu 1 (cu 2 )), 2(cu 1 )) = c(u 1 u 2, 2u 1 ) = ct (u) Both properties hold, therefore T is a linear transformation. Theorem 4 If T : R n R m is a linear transformation and e 1, e 2,..., e n are the the standard basis vectors for R n, then the standard matrix for T is [T = [ T (e 1 ) T (e 2 )... T (e n ) 11

12 Example 13 Let T be the orthogonal projection onto the yz-plane in R 3. Then 1 T (e 1 ) = T =, T (e 2 ) = T 1 = 1, T (e 3 ) = T 1 = 1 and therefore [T =