Question

use the definition of a linear operator to show that the function T:R^2 -> R^2 given the formula

T(x,y)= (x+2y , 3x-y) is a linear operator.

Answer 1

pls show step by step. esp for that one of T(u+v) = T(u) + T(v)

Answer 2

first of all check ur funtion u have T(x,y) =(x+2y,3x-y)

ur let
\[u=(x _{1},y_{1}) and v=(x_{2},y _{2})\]

from here
T(U+V)=
\[T(x _{1}+x _{2},y _{1}+y _{2})\]

if u r substituting u don't put T
by substituting i mean where u see x with subscript 1 or 2 u put (x+2y) with the corresponding subscript
where u see y with subscript 1or 2 u put (3x-y) with the corresponding subscript

now u'll have

\[(x _{1}+2y _{1}+x _{2}+2y _{2};3x _{1}-y _{1}+3x_{2}+y _{2})\]

separate them
\[(x _{1}-2y _{1};3x _{1}-y _{1}) + (x _{2}-2y _{2};3x _{2}-y _{2})\]

check this interesting part

\[(x _{1}+2y _{1};3x _{1}-y _{1}) =T(x _{1};y _{1})\]

but
\[(x _{1};y _{1})=u\]

then

u'll have
T(u) similarly do the same technique to v

Answer 3

the part wher ur separating.. what defn did u use to distribute + over ;