Is L(x,y) a Linear Transformation?

[aznkid310]

[SOLVED] Linear Transformation

Homework Statement

Determine if this is a linear transformation:

L(x,y) = (x+1, y, x+y)

Homework Equations

This is just one, but I have no clue as to how to even begin. I've been to lecture and read the book over and over again, but i was not given any relevant examples. Could someone please walk me through this? I know that to show it is a linear transformation, i must show that L(u+v) = L(u) + L(v), but i can't seem to find L(u+v)

The Attempt at a Solution

u =
[x
y]

v =
[x'
y']

L(u) + L(v) =
x + x' + 2
y + y'
x + y + x' +y'

I'm not even sure that is correct, but if it is, how does one find L(u+v)? Additionally, the fact that it is a transformation from R^2 => R^3 is throwing me off

 

[Dick]

The domain is R^2, they mean that (x,y)+(x',y') should be defined by (x+x',y+y'). What's L of that?

 

[aznkid310]

would L(x+x', y+y') = x+x'+2 ?
y+y'
x+1+y

It's probably extremely obvious, but i still don't understand.

 

[Dick]

L(x+x',y+y') would be (x+x'+1,y+y',x+x'+y+y'). Look at the definition. Substitute x+x' -> x and y+y' ->y. Notice that's different from what you found for L(u)+L(v).

 

[aznkid310]

ah that makes sense! I think i understand now.Thx for the help!