Relation between image(A) and image(A^2+A)

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SUMMARY

The discussion centers on the relationship between the image of a linear transformation A and the image of the transformation A^2 + A. It establishes that any vector x in the image of A^2 + A can be expressed as a linear combination of vectors in the image of A, specifically through the equation im(A^2 + A) = im(A^2 x + A x). The participants emphasize the need to formalize the approach by comparing the two images directly, suggesting a methodical examination of the elements in each image.

PREREQUISITES
  • Understanding of linear transformations and their properties
  • Familiarity with the concept of image in linear algebra
  • Knowledge of polynomial transformations of matrices
  • Basic skills in formal mathematical proof techniques
NEXT STEPS
  • Study the properties of linear combinations in vector spaces
  • Explore the implications of the Rank-Nullity Theorem
  • Learn about the relationship between images of polynomial transformations of matrices
  • Investigate examples of specific matrices to visualize the concepts discussed
USEFUL FOR

Students of linear algebra, mathematicians exploring linear transformations, and educators seeking to clarify the relationship between images of linear operators.

Candice
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Homework Statement


  1. What is the relation between the image of A and the image of A2 + A?

Homework Equations

The Attempt at a Solution



im (A^2 + A) for x (A^2+A) is within the image. Linear combination properties show A^2 x + A x. Not sure where to go from here
 
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Candice said:

Homework Statement


  1. What is the relation between the image of A and the image of A2 + A?

Homework Equations

The Attempt at a Solution



im (A^2 + A) for x (A^2+A) is within the image. Linear combination properties show A^2 x + A x. Not sure where to go from here

You need to start formalising your approach. You're really just writing down vague thoughts here!

You are asked to compare two sets. So, start by taking ##x \in Im(A)## and/or ##x \in Im(A^2 + A)## and see what you get.
 

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