Explain why the function f(x) = ex from R to R is not a linear transformation.
Solution
The function exp(x) is a exponential function , it is unbounded function i.e. for x= infinity this function is also infinity. Linear function should always satisfy the additivity and homoginity property. Consider the function y=f(x)=exp(x)-------------------(1) y1=exp(x1)--------------------------------------------------(2) y2=exp(x2)-------------------------------------------------(3) From eq. (1),put y=y1+y2 and x=x1+x2 So y1+y2=exp(x1+x2)-------------------------------------------(4) clearly eq.(4) is not equal to summation of eq.(2) and (3) so it will not satisfy the additivity property ,so this function is not linear transformation. Simply we can check homoginity f(ax)=exp(ax)----------------------------------------------------(5) now multiply (1) with a ,then y=a * exp(x)--------------------------------------(6) (5) not equal to (6) so it will not satisfy the homoginity property ,so this function is not linear transformation.
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Explain why the function f(x) - ex from R to R is not a linear transfo.docx
1. Explain why the function f(x) = ex from R to R is not a linear transformation.
Solution
The function exp(x) is a exponential function , it is unbounded function i.e. for x= infinity this
function is also infinity. Linear function should always satisfy the additivity and homoginity
property. Consider the function y=f(x)=exp(x)-------------------(1) y1=exp(x1)------------------------
--------------------------(2) y2=exp(x2)-------------------------------------------------(3) From eq.
(1),put y=y1+y2 and x=x1+x2 So y1+y2=exp(x1+x2)-------------------------------------------(4)
clearly eq.(4) is not equal to summation of eq.(2) and (3) so it will not satisfy the additivity
property ,so this function is not linear transformation. Simply we can check homoginity
f(ax)=exp(ax)----------------------------------------------------(5) now multiply (1) with a ,then y=a *
exp(x)--------------------------------------(6) (5) not equal to (6) so it will not satisfy the homoginity
property ,so this function is not linear transformation.
