Proving a transformation is linear

[_N3WTON_]

Homework Statement

If T is linear, show that it is linear by finding a standard matrix A for T so that:

Also show that this equation holds for the matrix you have found. If T is not linear, prove that T is not linear by showing that it does not fit the definition of a linear transformation

Homework Equations

Definition of a linear transformation:
$T(\vec{u}+\vec{v}) = T(\vec{u})+T(\vec{v})$
$T(c\vec{u}) = cT(\vec{u})$

The Attempt at a Solution

First I let $$ \vec{e}_{1} =
\begin{bmatrix}
1\\
0
\\0

\end{bmatrix} $$
$$ \vec{e}_{2} =
\begin{bmatrix}
0\\
1
\\0

\end{bmatrix}$$
$$ \vec{e}_{3} =
\begin{bmatrix}
0\\
0
\\1

\end{bmatrix}$$
However, when I go to separate
$$
\vec{b} = \begin{bmatrix}
2x_1 - 3x_2\\
2x_2\\
4x_1 + 3
\end{bmatrix}$$ I am not sure how to handle the constant, i.e, I am not sure how to rewrite as $A\vec{x}$. I think once I figure that out I should be able to do the rest of problem

 

[Orodruin]

It is not linear ... Just find a counter example to the definitions of what a linear transformation is.

 

[wabbit]

(Edit : deleted, duplicate response)

 

[_N3WTON_]

It is not linear ... Just find a counter example to the definitions of what a linear transformation is.

Ok, I didn't realize that was all I needed to do. I have attached a picture of my work because it would be kind of long to write out in latex:

 

[Dick]

You could simplify that a lot. If T is linear, then T(0)=0, since T(0x)=0T(x). Is it for your given T?

 

[_N3WTON_]

You could simplify that a lot. If T is linear, then T(0)=0, since T(0x)=0T(x). Is it for your given T?

wow I hadn't thought of that, sort of makes the problem trivial lol...anyway thanks for the help :)