Given a transformation of the plane F(x,y) = (2x+y,x-2y), find F^-1.

[kris89]

Homework Statement

Given a transformation of the plane F(x,y) = (2x+y,x-2y), find F^-1.

Homework Equations

Actually this exercise had an item (a) which I had to prove this is a transformation. So I proved this function is injective and surjective.

I know F(x,y) = (u,v) IFF F^-1(u,v) = (x,y).

The Attempt at a Solution

I did it, but I don't know if this is the real solution...

F(x,y) = (u,v) IFF F^-1(u,v) = (x,y).
u = 2x + y
v = x - 2y
Solving this, I concluded that: x = (v+2u)/5 and y = (u-2v)/5
Is this the solution? F^-1 = ((v+2u)/5, (u-2v)/5) or do I have to do anything else?

 

[Mark44]

Homework Statement

Given a transformation of the plane F(x,y) = (2x+y,x-2y), find F^-1.

Homework Equations

Actually this exercise had an item (a) which I had to prove this is a transformation. So I proved this function is injective and surjective.

I know F(x,y) = (u,v) IFF F^-1(u,v) = (x,y).

The Attempt at a Solution

I did it, but I don't know if this is the real solution...

F(x,y) = (u,v) IFF F^-1(u,v) = (x,y).
u = 2x + y
v = x - 2y
Solving this, I concluded that: x = (v+2u)/5 and y = (u-2v)/5
Is this the solution? F^-1 = ((v+2u)/5, (u-2v)/5) or do I have to do anything else?

Not quite. You should write the inverse as F^-1(u, v) = ((v+2u)/5, (u-2v)/5).

You've done most of the work already. All that's left is to show that F^-1(u, v) = (x, y). Replace u and v by what you have above, and use the formula for F^-1 that you found.

 

[HallsofIvy]

Alternatively, your teacher may prefer functions written in "x" and "y". In that case, replace your "u" with "x" and "v" with y: $F^{-1}(x, y)= ((y+ 2x)/5, (x- 2y)/5)$.