Linear Transformations Rn->Rm Question

[haribol]

Linear Transformations Rn-->Rm Question

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

Determine whether the following T:Rⁿ to R^m

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 what's happening step by step?



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

A transformation T:Rⁿ --> R^m 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 R^m

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 what's happening step by step?



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

A transformation T:Rⁿ --> R^m 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.