# Finding a Transformation Matrix to yield the basis

by silvermane
Tags: basis, matrices, solution space, transformation
 PF Gold P: 117 1. The problem statement, all variables and given/known data 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. 3. 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! :)))
 P: 351 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 $u'' = u$. 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?
 PF Gold P: 117 Well now... I feel like a noob :D e^t and e^-t are a basis... I misread the question -.-

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