Linear Transformations question

[johnnyboy2005]

I'm just wondering if someone can let me know if I'm on the right path here...

this question asks to show that the Function T: R^3 ----> R^2 given by the formula T(X1, X2, X3) = (2X1 - X2 + X3, X2 - 4X3) is a linear transformation.

soln' the definition of a L.T. is that T(u + v) = T(u) + T(V)

and T(cU) = cT(U)

so i show that T[(X1, X2, X3)+(Y1, Y2, Y3)] = T(X1, X2, X3) + T(Y1, Y2, Y3)...but what do i use as Y1, Y2, Y3? thanks for the time and help

 

[quasar987]

Just leave them like that. Using the definition of T, write out explicitely what T[(X1, X2, X3)+(Y1, Y2, Y3)] is, and write explicitely what T(X1, X2, X3) + T(Y1, Y2, Y3) is, and then compare.

 

[johnnyboy2005]

but what happens to (2X1 - X2 + X3, X2 - 4X3)?? do i just write the X's as Ys?

 

[HallsofIvy]

If f(x)= x², what is f(y)?

 

[johnnyboy2005]

y as in yes, i got it!