Section 4.3 Properties of Linear Transformations from R n to R m

ONE-TO-ONE TRANSFORMATIONS A linear transformation T: R n → R m is said to be one-to-one if T maps distinct vectors (points) in R n to distinct vectors (points) in R m.

THREE EQUIVALENT STATEMENTS Theorem 4.3.1: If A is an n×n matrix and T A : R n → R n is 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. NOTE: This extends our “big theorem” from Chapter 2. See Theorem on page 206.

INVERSE OF A 1-1 LINEAR OPERATOR If T A : R n → R n is a one-to-one linear operator, then [T] = A is invertible. The linear operator is called the inverse of T A. Notation: The inverse of T is denoted by T −1 and its standard matrix is [T −1 ] = [T] −1

PROPERTIES OF LINEAR TRANSFORMATIONS Theorem 4.3.2: A transformation T: R n → R m is linear if and only if the following relationships hold for all vectors u and v in R n and every scalar c. (a)T(u + v) = T(u) + T(v) (b)T(cu) = c T(u)

STANDARD BASIS VECTORS IN R n The standard basis vectors in R n are given by

A THEOREM Theorem 4.3.3: If T: R n → R m is a linear transformation, and e 1, e 2,..., e n are the standard basis vectors for R n, then the standard matrix for T is

EIGENVALUES AND EIGENVECTORS OF A LINEAR OPERATOR If T: R n → R n is a linear operator, then a scalar λ is called an eigenvalue of T if there is a nonzero x in R n such that T(x) = λx Those nonzero vectors x that satisfy this equation are called the eigenvectors of T corresponding to λ.