Check to see if the map is a linear transform 711 Which of t

Check to see if the map is a linear transform.

7.1.1. Which of the following functions F: R^3 rightarrow R are linear? (a) F(x, y, z)=x (b) F(x,y,z)=y-2 (c) F(x, y, z)=x+y+3 (d) F(x, y, z)=x?y?z (e) F(x, y, z) =xyz (f) F(x, y, z)=x^2-y^2+z^2 (g) F(x, y, z) = e^x-y+z Check to see if the map is a linear transform.

Solution

To check if a mapping is linear in general, all you need is verify the two properties.

The above two can be combined into one property: f(ax+by)=af(x)+bf(y)

a)f(ax+by)=ax+by=af(x,y,z)+bf(x,y,z)=aF(x)+bF(y)

Hence this is linear

b)f(ax+by)=ax+by-2=af(x,y,z)+bf(x,y,z)-2+2a+2b

Hence this is not linear

c)f(ax+by)=ax₁+by₁+3+ax₂+by₂+3=af(x₁,x₂,x₃)+bf(y₁,y₂,y₃)+6-3a-3b

Hence this is not linear

d)f(ax+by)=ax₁+by₁-ax₂-by₂-ax₃-by₃=ax₁-ax₂-ax₃₊by₁-by₂-by₃₌af(x₁,x₂,x₃)+bf(y₁,y₂,y₃)

Hence this is linear.

You can do the rest its similar. you can also do it by checking each property .