# Proving a transformation is linear

1. Feb 28, 2015

### _N3WTON_

1. The problem statement, all variables and given/known data

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

3. 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

2. Feb 28, 2015

### Orodruin

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

3. Feb 28, 2015

### wabbit

(Edit : deleted, duplicate response)

Last edited: Feb 28, 2015
4. Feb 28, 2015

### _N3WTON_

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:

5. Feb 28, 2015

### 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?

6. Feb 28, 2015

### _N3WTON_

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