Question

can you please show me step by step how to answer this 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

1) \(f(\alpha x)=\alpha f(x)\)
2) \(f(x_1+x_2)=f(x_1)+f(x_2)\)

Answer 2

yes, the above two are conditions that are required for me to conclude that the function T is a linear function right? but how do i use it?

Answer 3

In the first case replace \((x,y)\) with \(\alpha(x,y)=(\alpha x, \alpha y)\) and see what you will get.

Answer 4

All you need to know is how to multiply a scalar on a vector and how to add two vectors.

Answer 5

T(αx,αy) =α(x+2y),α(3x-y)
=T α( x+2y, 3x-y)

Answer 6

It is 0:08 a.m. And I am tired. Please, reply faster if you want to get an answer.

Answer 7

i'm sorry. is the above correct for the first condition?

Answer 8

This is the full answer:
T(αx,αy) =(αx+2αy,3αx-αy) =(α(x+2y),α(3x-y))=α(x+2y,3x-y)=αT(x,y)

Answer 9

The next step looks like this. Replace \((x,y)\) with \((x_1+x_2,y_1+y_2)\) and see what you'll get.

Answer 10

It is too late. Bye-bye.

Answer 11

thank you. im following on the steps

Answer 12

T(x1+x2,y1+y2) = (x1+x2)+2(y1+y2),3(x1+x2)-(y1+y2)
= x1+x2+2y1+2y2 , 3x1+3x2-y1-y2
= (x1+2y1)+(x2+2y2) , (3x1-y1)+(3x2-y2)

if i'm correct thus far, how do i continue?