Finding a Transformation Matrix to yield the basis

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SUMMARY

The discussion centers on solving the linear differential equation d²u/dt² = u within a vector space. The key solutions identified are e^t and e^-t, which form a basis for the solution space. The initial confusion regarding the need for a Transformation Matrix was clarified, emphasizing that the goal is to express any vector that solves the equation as a linear combination of these two basis vectors.

PREREQUISITES
  • Understanding of linear differential equations
  • Familiarity with vector spaces and basis concepts
  • Knowledge of exponential functions and their properties
  • Basic skills in solving second-order differential equations
NEXT STEPS
  • Study the theory of linear differential equations, focusing on second-order equations
  • Learn about the concept of basis in vector spaces and its applications
  • Explore the method of finding solutions to differential equations using exponential functions
  • Investigate the role of Transformation Matrices in linear algebra
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Students studying differential equations, mathematicians exploring vector spaces, and educators teaching linear algebra concepts.

silvermane
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Homework Statement


The solutions to the linear differential equation d^2u/dt^2 = u for a vector space. Find two independent solutions, to give a basis for that solution space.

The Attempt at a Solution


I want to understand this question. I feel that there's something I'm missing. I believe that I need to find the Transformation Matrix, but I need to understand how :(

Any hints or tips are greatly appreciated, but please don't give me just an answer! :)))
 
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I don't really think you're looking for a transformation matrix here. You have some vector space, with a vector u. There are a bunch of vectors (an infinite number, in fact!) in the space that solve the equation [itex]u'' = u[/itex]. What you want is a way to write down any vector that solves that equation as the sum of two other vectors (that you are going to find).

Now, what vectors solve this equation?
 
Well now... I feel like a noob :D

e^t and e^-t are a basis... I misread the question -.-
 

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