Please help with the following:?

Well,

Let F : R^2 -> R^3 be given by
F(x; y) = (x + 2y, y, x - y).
Prove that this is a linear transformation using the definition of a linear transformation.

in this case, u and v are vectors of R^2, that is, ordered pairs.
i)
let u1 = (x1, y1) and u2 = (x2, y2) be any two vectors of R^2
then
F(u1 + u2) = F( x1+x2, y1+y2)
= ( (x1+x2) + 2(y1+y2), (y1+y2) , (x1+x2) - (y1+y2) )
= ( (x1+2y1) + (x2+2y2), y1+y2, (x1 - y1) + (x2 - y2) )
= (x1 + 2y1, y1, x1 - y1) + (x2 + 2y2, y2, x2 - y2)
= F(x1, y1) + F(x2, y2)
= F(u1) + F(u2)
therefore : property (i) is verified

ii) let u = (x, y) be any vector of R^2 and k be any scalar (real number)
then
F(ku) = F( k * (x, y) )
= F( (kx, ky) )
= ( (kx) + 2(ky), ky, kx - ky)
= k * (x + 2y, y, x - y)
= k * F(u)
therefore : property (ii) is verified

conclusion :
we proved that F is a linear transformation from R^2 to R^3

qed

hope it' ll help !!

PS: if you want good answers, don't forget to give BAs too !! ;-)

OK. Where are you stuck?
i) Evaluate F(u). Evaluate F(v). Evaluate F(u + v). Compare. What is your question on this step?
ii) Evaluate F(ku). Evaluate k * F(u). Compare. What is your question on this step?

It's just plugging variables and expressions into a function. What is your question?