How do I show if T(x,y)=(2x+y,x-y) is linear?

Question

anonymous · Answer

Show that T(av + u) = aT(v) + T(u)

anonymous · Answer

Do you understand what to do?

anonymous · Answer

interested in this ...

anonymous · Answer

\begin{eqnarray*}
T((ax_1, ay_1) + (x_2, y_2)) &=&
T(ax_1 + x_2, ay_1 + y_2)\ &=& 
(2(ax_1 + x_2) + (ay_1 + y_2), (ax_1 + x_2) - (ay_1 + y_2)) \ &=& 
(2ax_1 + 2x_2 + ay_1 + y_2, ax_1 + x_2 - ay_1 - y_2) \ &=&
(a(2x_1 + y_1) + (2x_2 + y_2), a(x_1 - y_1) + (x_2 - y_2)) \ &=&
(a(2x_1 + y_1),  a(x_1 - y_1)) + (2x_2 + y_2, x_2 - y_2) \ &=&
a(2x_1 + y_1, x_1 - y_1) + (2x_2 + y_2, x_2 - y_2) \ &=&
aT(x_1, y_1) + T(x_2, y_2)
\end{eqnarray*}

anonymous · Answer

Therefore T is a linear transformation.

anonymous · Answer

Yeah it make sense now, I got the same thing, thanks!

myininaya · Answer

awesome job alchemista :)