10/4/2003, LINEAR TRANSFORMATIONS Math 21b, O. Knill HOMEWORK. For Wednesday: Section 2.1: 6,14,28,42,44,34*,(24-30)* TRANSFORMATIONS. A transformation T from a set X to a set Y is a rule, which assigns to every element in X an element y = T ( x ) in Y . One calls X the domain and Y the codomain . A transformation is also called a map . LINEAR TRANSFORMATION. A map T from R n to R m is called a linear transformation if there is a m × n matrix A such that T ( ~x ) = A~x . EXAMPLES. • To the linear transformation T ( x, y ) = (3 x +4 y, x +5 y ) belongs the matrix ± 3 4 1 5 ² . This transformation maps the plane onto itself. • T ( x ) =-3 x is a linear transformation from the real line onto itself. The matrix is A = [-3]. • To T ( ~x ) = ~ y · ~x from R 3 to R belongs the matrix A = ~ y = ³ y 1 y 2 y 3 ´ . This 1 × 3 matrix is also called a row vector . If the codomain is the real axes, one calls the map also a function . function de±ned on space. • T ( x ) = x~ y from R to R 3 . A = ~ y = y 1 y 2 y 3 is a 3 × 1 matrix which is also called a column vector . The map de±nes a line in space. • T
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This note was uploaded on 04/06/2008 for the course MATH 21B taught by Professor Judson during the Spring '03 term at Harvard.