Show that T is a nonlinear transformation

In summary, the given function T is not a linear transformation because it does not satisfy the properties of linearity, specifically additivity and homogeneity. A counterexample is given by choosing the vector v = [0 0]^T, where T(v) = [-1 1]^T, which violates the property of additivity. This is further supported by the fact that T applied to the vector [-1 1]^T does not yield the same result as the original vector, indicating a nonlinear transformation.
  • #1
Sociomath
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}##.
 
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  • #2
Sociomath said:
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?
 

1. What is a nonlinear transformation?

A nonlinear transformation is a mathematical operation that does not follow a straight or linear relationship between the input and output. This means that the change in the output is not proportional to the change in the input.

2. How do you show that T is a nonlinear transformation?

To show that T is a nonlinear transformation, you can use the property of superposition, which states that if T is a linear transformation, then T(a + b) = T(a) + T(b) for any input vectors a and b. If this property does not hold, then T is a nonlinear transformation.

3. What are some examples of nonlinear transformations?

Examples of nonlinear transformations include logarithmic functions, exponential functions, and trigonometric functions. These functions do not have a constant rate of change and their graphs are curved.

4. Why is it important to identify a transformation as nonlinear?

Identifying a transformation as nonlinear is important because it allows us to understand its behavior and properties better. Nonlinear transformations have unique characteristics that differ from linear transformations, and knowing this distinction can help us solve mathematical problems more accurately.

5. Can a transformation be both linear and nonlinear?

No, a transformation cannot be both linear and nonlinear. A transformation is either one or the other, depending on whether or not it follows the property of superposition. If it does, then it is linear, and if it does not, then it is nonlinear.

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