1 Give the definition of a linear function For each of the f

1. Give the definition of a linear function. For each of the following functions, decide if the given function is linear, justifying your answer in each case.

(a) f : R R, where f(x) = x,

(b) g : R 3 R 2 , where g(x, y, z) = (x, y).

(c) h : R 2 R 2 , where h(x, y) = (0, x + y),

Solution

A linear function is a polynomial function of degree zero or one, or is the zero polynomial. A linear function, usually, has one independent variable and one dependent variable. When the function is of only one variable, it is of the form f(x) = ax + b, where a and b are arbitrary constants.

In linear algebra, a linear function f : Rⁿ R^m is a called a linear function if it satisfies the

following two properties:

(1) f(x + y) = f(x) + f(y), for all x, y Rⁿ

(2) f(cx) = cf(x) for x Rⁿ and c R

(a) The degree of the function f(x) = x is ½. Hence it is not a linear function.

(b) g: R³R² is defined by g(x ,y, z) = (x ,y). Then g[(x₁ ,y₁,z₁)+(x₂ ,y₂ ,z₂) ] = g(x₁ +x₂ ,y₁ +y₂ ,z₁ +z₂) = (x₁ +x₂, y₁+y₂) = g(x₁,y₁,z₁) + g(x₂,y₂,z₂). Further, g[ c(x, y, z)] = g(cx, cy, cz)= (cx,cy) = c(x,y) = c g(x,y,z).Thus, g satisfies both the requirements of a linear map. Hence g is a linear function.

(c ) h: R² R² is defined by h9x,y) = (0,x+y). Then h[(x₁,y₁)+(x₂,y₂]) =h(x₁+x₂,y₁+y₂) = (0, x₁+x₂+y₁+y₂) = (0, x₁+y₁+x₂+y₂) = (0, x₁+y₁) +(0, x₂+y₂ ) = h(x₁, y₁) +h(x₂,y₂). Further, h[c(x,y)] = h(cx,cy)= (0,cx+cy) = c(0,x+y) = c h(x,y). Thus, h satisfies both the requirements of a linear map. Hence h is a linear function.