A little help with Linear Transformations

In summary, the conversation is about the speaker struggling with using mathematical proof in linear algebra and asking for help with proving that a given transformation is not linear. The speaker acknowledges the steps needed to prove linearity, but is unsure where to begin. They also mention the concept of linear transformations and the importance of having first degree formulas.
  • #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
Reading my book I know that I need to prove that the transformation is closed under additivity and scalar multiplication, ...

That's what you would do if you wanted to prove that T /was/ a linear transformation. You're asked to prove T is not a linear transformation. You can do this by exhibiting two vectors a, b such that T(a + b) != T(a) + T(b), for example.
 
  • #3
look, the whole idea of a, onear transformation is that when it is given in formulas, the formulas are linear, i.e. first degree. what degree are your formulas?
 

Related to A little help with Linear Transformations

1. What is a linear transformation?

A linear transformation is a function that maps one vector space to another, while preserving the basic algebraic structure of the original space. In other words, it is a function that takes in a vector and produces a new vector in the same space.

2. How do I know if a transformation is linear?

A transformation is considered linear if it satisfies two properties:

  1. Additivity - T(u + v) = T(u) + T(v)
  2. Homogeneity - T(ku) = kT(u)
where u and v are vectors, and k is a scalar. If these properties hold, then the transformation is linear.

3. What is the importance of linear transformations?

Linear transformations are important in various fields, such as mathematics, physics, and engineering. They allow us to study and understand complex systems by breaking them down into simpler components. They also have many practical applications, such as image and signal processing, data compression, and computer graphics.

4. Can a linear transformation change the dimension of a vector space?

Yes, a linear transformation can change the dimension of a vector space. For example, a transformation that maps a 3-dimensional space to a 2-dimensional space will change the dimension of the original space. However, the dimension of the transformed space will still be preserved, meaning that the number of basis vectors needed to span the space will remain the same.

5. How are matrices related to linear transformations?

Matrices and linear transformations are closely related, as matrices can be used to represent linear transformations. Each column of a matrix represents the image of a basis vector under the transformation. Additionally, matrix multiplication can be used to compute the result of a linear transformation on a given vector.

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