6. Linear Transformations

Home , Invertible matrix, Projection (mathematics)

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: RnRn, 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:RnRm 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:R3R2 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:R2R2 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:R2R2 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.

113 1 221   4 4 33  11.18 P /12 x   111  3 10.321 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.