Show that T is a nonlinear transformation

  • Thread starter Sociomath
  • Start date
  • #1
9
0
1. Show that T isn't a linear transformation and provide a suitable counterexample.
##T \begin{bmatrix}x\\y \end{bmatrix} = \begin{bmatrix}x - 1 \\ y + 1 \end{bmatrix}##

2. The attempt at a solution
##\text{let}\, \vec{v} = \begin{bmatrix}0\\0 \end{bmatrix}. \text{Then,}##
##T(\vec{v}) = T\left(\begin{bmatrix}0\\0 \end{bmatrix}\right) = \begin{bmatrix}0 - 1 \\ 0 + 1 \end{bmatrix} = \begin{bmatrix}-1 \\ 1 \end{bmatrix}##
Nonlinear transformation due to constants: ##T \begin{bmatrix}-1 \\ 1 \end{bmatrix}##.
 

Answers and Replies

  • #2
36,013
7,944
1. Show that T isn't a linear transformation and provide a suitable counterexample.
##T \begin{bmatrix}x\\y \end{bmatrix} = \begin{bmatrix}x - 1 \\ y + 1 \end{bmatrix}##

2. The attempt at a solution
##\text{let}\, \vec{v} = \begin{bmatrix}0\\0 \end{bmatrix}. \text{Then,}##
##T(\vec{v}) = T\left(\begin{bmatrix}0\\0 \end{bmatrix}\right) = \begin{bmatrix}0 - 1 \\ 0 + 1 \end{bmatrix} = \begin{bmatrix}-1 \\ 1 \end{bmatrix}##
Why does the above show that T is not a linear transformation?
And why is ## \begin{bmatrix}0\\0 \end{bmatrix}## a counter example?
Sociomath said:
Nonlinear transformation due to constants: ##T \begin{bmatrix}-1 \\ 1 \end{bmatrix}##.
Your reason doesn't make sense to me, and this part -- ##T \begin{bmatrix}-1 \\ 1 \end{bmatrix}## -- really doesn't make sense.

What are the properties a linear transformation has to satisfy?
 

Related Threads on Show that T is a nonlinear transformation

Replies
8
Views
2K
Replies
2
Views
556
Replies
6
Views
2K
  • Last Post
Replies
1
Views
2K
Replies
2
Views
874
Replies
16
Views
382
Replies
15
Views
2K
Replies
4
Views
1K
Top