Linear Transformations Rn-->Rm Question

[haribol]

I would be very grateful if someone can explain what is going on in the following problem:

Determine whether the following T:Rn to Rm

T(x,y)=(2x,y)

Solution from solutions manual:

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

My questions are

1. Where did the x1's and the x2's and the y1's and the y2's come from?

2. Can you please explain whats happening step by step?



[PS]--> The questions asks to use the theorem which states:

A transformation T:Rn --> Rm is linear if and only if the following relationships hold for all vectors u and v in Rn and every scalar c

a) T(u+v) = T(u) + T(v)

b)T(cu) = cT(u)

[quasar987]

I would be very grateful if someone can explain what is going on in the following problem:

Determine whether the following T:Rn to Rm

T(x,y)=(2x,y)

Solution from solutions manual:

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

My questions are

1. Where did the x1's and the x2's and the y1's and the y2's come from?

2. Can you please explain whats happening step by step?



[PS]--> The questions asks to use the theorem which states:

A transformation T:Rn --> Rm is linear if and only if the following relationships hold for all vectors u and v in Rn and every scalar c

a) T(u+v) = T(u) + T(v)

b)T(cu) = cT(u)

He has set \vec{u} = (x_1,y_1), \ \ \vec{v} = (x_2,y_2) and showed using vector addition properties that T(\vec{u}+\vec{v}) = T(\vec{u})+T(\vec{v})
This proof is imcomplete though because he left out condition b).

[haribol]

Thank you quasar987 for the clarification. The manual does include the proof using condition b) but I forgot to type it.

Thanks for that clarification.