Determine if the function is a linear tranformation.

Question

Is f a linear transformation?

Let f: R^2 --> R be defined by f(<x,y>) = 2x - 9y. Is f a linear transformation?

a. f (<x1,y1>) + <x2,y2>) = ?

f (<x1,y1) + f (<x2,y2>) = ?

b. f (c<x,y>) = ?

c (f (<x,y>)) = ?

c. Is f a linear transformation?

Answer 1

The idea is showing it is closed under addition and scalar multiplication.

f(<x,y>) = 2x-9y

f(<x_1,y₁>+<x₂,y₂>)= f(<x₁+x₂, y₁+y₂>=2(x₁+x₂)-9(y₁+y₂)=2x₁-9y₁+2x₂-9y₂=f(<x₁,y₁>)+f(<x₂,y₂>)

You can verify this step and inlude any steps I may have skipped over.

You can also use the same process to show that the identity (origin) of R² maps to the identity (zero) of R

and the same for the scalar multiplication. Thus you can make your conculusion

Answer 2

The idea is showing it is closed under addition and scalar multiplication.

f(<x,y>) = 2x-9y

f(<x_1,y₁>+<x₂,y₂>)= f(<x₁+x₂, y₁+y₂>=2(x₁+x₂)-9(y₁+y₂)=2x₁-9y₁+2x₂-9y₂=f(<x₁,y₁>)+f(<x₂,y₂>)

You can verify this step and inlude any steps I may have skipped over.

You can also use the same process to show that the identity (origin) of R² maps to the identity (zero) of R

and the same for the scalar multiplication. Thus you can make your conculusion