How to Solve Linear Transformations with Only a Constant?

[EV33]

Homework Statement

F:R^2 to R^2 defined by

F(x)=
x1+x2
1


Where x=
x1
x2

Homework Equations

Must satisfy these conditions:
T(u+v)=T(u)+T(v)
T(au)=aT(u)

The Attempt at a Solution

I said
u=
u1
u2

v=
v1
v2

u+v=
u1+u2
v1+v2

then F(u+v)=
(u1+v1) + (u2+v2)
...

This is where I got confused.

Because there is only a constant in the bottom row, which is a 1,
does this mean it is not a transformation? I don't know how to solve these when there is only a constant.

 

[n!kofeyn]

You're exactly right. You have
F(u+v) = \begin{pmatrix}<br /> u_1+v_1+u_2+v_2 \\<br /> 1<br /> \end{pmatrix} \not= \begin{pmatrix}<br /> u_1+u_2 \\<br /> 1<br /> \end{pmatrix} + \begin{pmatrix}<br /> v_1+v_2 \\<br /> 1<br /> \end{pmatrix} = F(u)+F(v)
So F is not a linear transformation.

 

[Dick]

You are doing fine. Yes, there is a 1 in the second row. Just write it down. Now is F(u+v)=F(u)+F(v)?

 

[EV33]

No it is not. Thank you very much.