Question

I WANT TO ASK ABOUT LINEAR TRANSFORMATION

Answer 1

what u wanna ask?

Answer 2

if the question like this L(x,y)=(x+y,x-y)
how can i know that it is a linear transformation???

Answer 3

A transformation \(L: V \rightarrow W\) is said to be linear if it has the following two properties:
1) \(L(v+u)=L(v)+L(u) \text{, u and v are two vectors in V.}\)
2) \(L(cv)=cL(v) \text{, v is a vector in V and c is a constant.}\)

Answer 4

Now let's see weather the transformation we have is linear or not by testing it to each property.
Let \((x_1,y_1)\in v \text{ and } (x_2,y_2)\in u\), then:
1) \(L(u+v)=L(x_1+x_2,y_1+y_2)=(x_1+x_2+y_1+y_2,x_1+x_2-(y_1+y_2))\)
\(=(x_1+y_1,x_1-y_1)+(x_2+y_2,x_2-y_2)=L(v)+L(u).\)
2) \(L(cv)=L(cx_1,cy_1)=(cx_1+cy_1,cx1-cy_1)=c(x_1+y_1,x_1-y_1)=cL(v).\)
We can see that the it satisfies both properties and therefor \(L\) is a linear transformation.

Answer 5

I meant: let \(v=(x_1,y_1) \text{ and } u=(x_2,y_2)\). That doesn't affect the answer anyway.