- #1

haribol

- 52

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**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:R

^{n}to R

^{m}

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

Solution from solutions manual:

T((x

_{1},y

_{1}) + (x

_{2},y

_{2})) = (2(x

_{1}+x

_{2}), y

_{1}+y

_{2}) = (2x

_{1},y

_{1}) + (2x

_{2},y

_{2}) = T(x

_{1},y

_{1}) + T(x

_{2},y

_{2})

*My questions are*

1. Where did the x

_{1}'s and the x

_{2}'s and the y

_{1}'s and the y

_{2}'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:R

^{n}--> R

^{m}is linear if and only if the following relationships hold for all vectors

**u**and

**v**in R

^{n}and every scalar c

a) T(

**u**+

**v**) = T(

**u**) + T(

**v**)

b)T(c

**u**) = cT(

**u**)