What are the effects of a translation on a vector in R2?

[Mdhiggenz]

Homework Statement

Let a be a fixed nonzero vector in R2. A mapping of the form L(x) = x + a is called a
translation. Show that a translation is not a linear transformation. Illustrate geometrically the effect of a translation.

My work is in the photo below, can you check and see if I'm correct.

Also to show geometrically the effect of a translation, I just drew a vector being shifted a units.

Thanks

Higgenz

Homework Equations

The Attempt at a Solution

 

[Fredrik]

You should try to type your posts instead of posting a picture. Typed text is so much easier to read. If you want to type a matrix, you can do it like this:
$$\begin{bmatrix}a & b\\ c & d\end{bmatrix}$$ Hit the quote button next to my post to see how I did this. More information here.

The statement of the problem is a bit careless. There is a translation that's linear: the one with a=0. The problem should be asking you to prove one of the following statements:

1. If L is a translation by $a$ and $a\neq 0$, then L is not linear.
2. If L is a translation by $a$ and L is linear, then a=0.

The calculation you're doing is fine, but since you're just showing the calculation, and not including any "for all" or "there exists" statements, it's not clear what you're proving. If you decide to prove statement 1 above, you need to think about what exactly "L is not linear" means.

Also, I think you should change the notation so that the components of u are denoted by $u_1$ and $u_2$, and similarly for the other variables.