Proving Non-Linearity of a Transformation in R^3

In summary, the conversation discusses the difficulty of using mathematical proof in Linear Algebra and the need to prove that a given transformation is not linear. The suggested approach is to show a counter example by applying the transformation to specific vectors.
  • #1
Rounder01
2
0
Okay, I will just admit that I stink at using mathematical proof in Linear. I hope someone can give me a push with this problem

Prove that T : R(real)^3 -> R(real)^3 defined by T([yz,xz,zy]) is not a linear transformation.

Reading my book I know that I need to prove that the transformation is closed under additivity and scalar multiplication, but alas I do not know where to begin with this. Any help would be appreciated.
 
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  • #2
I never did any proofs in linear algebra, but I think you can prove it by showing counter example.
linear transformation means T(v+w) = T(v) + T(w) where v,w are vectors in R^3
take v=[1 0 0] and w=[0 0 1] and see what you can when you apply transformation.
 

FAQ: Proving Non-Linearity of a Transformation in R^3

What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another in a way that preserves the fundamental operations of vector addition and scalar multiplication.

What is the difference between a linear transformation and a nonlinear transformation?

A linear transformation follows the rules of linearity, meaning that the output is directly proportional to the input. In contrast, a nonlinear transformation does not follow these rules and can have more complex relationships between the input and output.

How do you represent a linear transformation mathematically?

A linear transformation can be represented by a matrix. The input vector is multiplied by the transformation matrix to produce the output vector. The matrix must have the same number of columns as the input vector and the same number of rows as the output vector.

What is the significance of eigenvalues and eigenvectors in linear transformations?

Eigenvalues and eigenvectors are important in linear transformations because they represent the scaling and direction of the transformation. Eigenvalues determine the amount of scaling, while eigenvectors determine the direction of the scaling.

How are linear transformations useful in real-world applications?

Linear transformations are useful in many fields such as computer graphics, physics, and economics. They can be used to represent and manipulate data, model real-world systems, and solve complex problems in a more efficient and organized manner.

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