6.6. LinearLinear TransformationsTransformations E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.1.6.1. MatricesMatrices asas TransformationsTransformations A Review of Functions domain codomain range x y preimage image http://en.wikipedia.org/wiki/Codomain E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.1.6.1. MatricesMatrices asas TransformationsTransformations A Review of Functions A function whose input and outputs are vectors is called a transformation, and it is standard to denote transformations by capital letters such as F, T, or L. w=T(x) “T maps x into w” E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.1.6.1. MatricesMatrices asas TransformationsTransformations A Review of Functions If T is a transformation whose domain is Rn and whose range is in Rm, then we will write (read, “T maps Rn into Rm”). You can think of a transformation T as mapping points into points or vectors into vectors. If T: RnRn, then we refer to the transformation T as an operator on Rn to emphasize that it maps Rn back into Rn. E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.1.6.1. MatricesMatrices asas TransformationsTransformations Matrix Transformations Matrix transformation If A is an m×n matrix, and if x is a column vector in Rn, then the product Ax is a vector in Rm, so multiplying x by A creates a transformation that maps vectors in Rn into vectors in Rm. We call this transformation multiplication by A or the transformation A and denote it by TA to emphasize the matrix A. and or equivalently, nn In the special case where A is square, say n×n, we have TR A : R , n and we call TA a matrix operator on R . E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.1.6.1. MatricesMatrices asas TransformationsTransformations Linear Transformations The operational interpretation of linearity 1. Homogeneity: Changing the input by a multiplicative factor changes the output by the same factor; that is, 2. Additivity: Adding two inputs adds the corresponding outputs; that is, E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.1.6.1. MatricesMatrices asas TransformationsTransformations Linear Transformations n If v1, v2, …, vk are vectors in R and c1, c2, …, ck are any scalars, then Engineers and physicists sometimes call this the superposition principle. E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.1.6.1. MatricesMatrices asas TransformationsTransformations Linear Transformations Example 7 From Theorem 3.1.5, If A is an m×n matrix, u and v are column vectors in Rn, and c is a scalar, then A(cu)=c(Au) and A(u+v)=Au+Av. n m Thus, the matrix transformation TA:R R is linear since E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.1.6.1. MatricesMatrices asas TransformationsTransformations Some Properties of Linear Transformations E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.1.6.1. MatricesMatrices asas TransformationsTransformations All Linear Transformations from Rn to Rm Are Matrix Transformations The matrix A in this theorem is called the standard matrix for T, and we say that T is the transformation corresponding to A, or that T is the transformation represented by A, or sometimes simply that T is the transformation A. E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.1.6.1. MatricesMatrices asas TransformationsTransformations All Linear Transformations from Rn to Rm Are Matrix Transformations E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.1.6.1. MatricesMatrices asas TransformationsTransformations All Linear Transformations from Rn to Rm Are Matrix Transformations When it is desirable to emphasize the relationship between T and its standard matrix, we will denote A by [T]; that is, we will write With this notation, the relation ship in (13) becomes (14) (13) E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.1.6.1. MatricesMatrices asas TransformationsTransformations All Linear Transformations from Rn to Rm Are Matrix Transformations REMARK Theorem 6.1.4 shows that a linear transformation T:RnRm is completely determined by its values at the standard unit vectors in the sense that once the images of the standard unit vectors are known, the standard matrix [T] can be constructed and then used to compute images of all other vectors using (14) Example 11 Show that the transformation T:R3R2 defined by the formula is linear and find its standard matrix. E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.1.6.1. MatricesMatrices asas TransformationsTransformations Rotations About The Origin Let θ be a fixed angle, and consider the operator T that rotates each vector x in R2 about the origin through the angle θ. T is linear. E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.1.6.1. MatricesMatrices asas TransformationsTransformations Rotations About The Origin We will denote the standard matrix for the rotation about the origin through an angle θ by Rθ. E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.1.6.1. MatricesMatrices asas TransformationsTransformations Reflections About Lines Through The Origin Let us consider the operator T:R2R2 that reflects each vector x about a line through the origin that makes an angle θ with the positive x-axis. E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.1.6.1. MatricesMatrices asas TransformationsTransformations Reflections About Lines Through The Origin E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.1.6.1. MatricesMatrices asas TransformationsTransformations Reflections About Lines Through The Origin E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.1.6.1. MatricesMatrices asas TransformationsTransformations Orthogonal Projections onto Lines Through The Origin Consider the operator T:R2R2 that projects each vector x in R2 onto a line through the origin by dropping a perpendicular to that line. E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.1.6.1. MatricesMatrices asas TransformationsTransformations Orthogonal Projections onto Lines Through The Origin The standard matrix for an orthogonal projection onto a general line through the origin can be obtained using Theorem 6.1.4. Consider a line through the origin that makes an angle θ with the positive x-axis, and denote the standard matrix for the orthogonal projection by Pθ. Solving for Pθx yeilds so part (b) of Theorem 3.4.4 implies that E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.1.6.1. MatricesMatrices asas TransformationsTransformations Orthogonal Projections onto Lines Through The Origin Example 14 Find the orthogonal projectin of the vector x=(1,1) on the line through the origin that makes an angle of π/12(=15º) with the x-axis. 111 3 1 33 22 4 11.184 P /12 x 111 3 10.321 33 422 4 E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.1.6.1. MatricesMatrices asas TransformationsTransformations Orthogonal Projections onto Lines Through The Origin E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.1.6.1. MatricesMatrices asas TransformationsTransformations Transformations of The Unit Square 2 The unit square in R is the square that has e1 and e2 as adjacent side; its vertices are (0,0), (1,0), (1,1), and (0,1). It is often possible to gain some insight into the geometric behavior of a linear operator on R2 by graphing the images of these vertices. E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.2.6.2. GeometryGeometry ofof LinearLinear OperatorsOperators Norm-Preserving Linear Operators Length preserving, angle preserving A linear operator T:RnRn with the length-preserving property ||T(x)||=||x|| is called an orthogonal operator or a linear isometry (from the Greek isometros, meaning “equal measure”). E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.2.6.2. GeometryGeometry ofof LinearLinear OperatorsOperators Norm-Preserving Linear Operators (a)(b) Suppose that T is length preserving, and let x and y be any two vectors in Rn. (4) E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.2.6.2. GeometryGeometry ofof LinearLinear OperatorsOperators Norm-Preserving Linear Operators (b)(a) Conversely, suppose that T is dot product preserving, and let x be any vector in Rn. Since It follows that E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.2.6.2. GeometryGeometry ofof LinearLinear OperatorsOperators Orthogonal Operators Preserve Angles And Orthogonality Recall from the remark following Theorem 1.2.12 that the angle between two nonzero vectors x and y in Rn is given by the formula Thus, if T:RnRn is an orthogonal operator, the fact that T is length preserving and dot product preserving implies that which implies that an orthogonal operator preserves angles. E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.2.6.2. GeometryGeometry ofof LinearLinear OperatorsOperators Orthogonal Matrices Our next goal is to explore the relationship between the orthogonality of an operator and properties of its standard matrix. Suppose that A is the standard matrix for an orthogonal linear operator T:RnRn. Since T(x)=Ax for all x in Rn, and since ||T(x)||=||x||, it follows that for all x in Rn. Axx22 AAxxxx AT AI A1 AT T AAxxxx T xxxxTTAA T E-mail: [email protected] http://web.yonsei.ac.kr/hgjung 6.2.6.2.