Struggling with Linear Transformation Part Two?

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SUMMARY

This discussion focuses on the challenges of understanding linear transformations, specifically in determining the similarity between matrices. The key takeaway is the process of finding a matrix S such that D = S-1AS, where A is a matrix representing a linear map T. It emphasizes the importance of the characteristic polynomial of A, as its roots and associated vectors are crucial for constructing the matrix S. Understanding the conditions under which matrices are diagonalizable is also highlighted as essential knowledge.

PREREQUISITES
  • Linear algebra concepts, specifically matrix diagonalization
  • Understanding of linear transformations and their representations
  • Familiarity with characteristic polynomials
  • Knowledge of equivalence relations in matrix theory
NEXT STEPS
  • Study the properties of diagonalizable matrices in linear algebra
  • Learn how to compute the characteristic polynomial of a matrix
  • Explore the concept of matrix similarity and its implications
  • Practice finding eigenvalues and eigenvectors for various matrices
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as educators seeking to clarify concepts related to linear transformations and matrix theory.

mslodyczka
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Hi,
I'm having trouble with part two of this question. If anyone can help me out with this I would appreciate it. Thanks,
Mike
 

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mslodyczka said:
Hi,
I'm having trouble with part two of this question. If anyone can help me out with this I would appreciate it. Thanks,
Mike


Some suggetions:

Understand why not all matrices are diagonalizable.

Let's assume this one is and let's call it A (i.e., A is the
representation of linear map T relative to standard basis E).

Let diagonal matrix D represent T relative to basis B.

Task: Find S such that D = S^-1AS. The columns of S are
the members of B. Done.

(Note This is an equivalence relation on matrices.
Matrices A and D are *similar*.
This should give some guidance in answering question iii.)

How to find S?
Write out the characteristic polynomial equation for A.
Solve it. The roots are key. The vectors associated with
these roots will make up the columns of S.
 

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