Finding a Transformation Matrix to yield the basis

In summary, the conversation discusses the process of finding solutions to a linear differential equation and the need to find two independent solutions to create a basis for the solution space. The conversation also clarifies that the solution involves finding vectors that satisfy the equation u'' = u and that the basis for the solution space includes e^t and e^-t.
  • #1
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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  • #2
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?
 
  • #3
Well now... I feel like a noob :D

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

Related to Finding a Transformation Matrix to yield the basis

What is a transformation matrix?

A transformation matrix is a mathematical representation of a transformation, which involves changing the position, orientation, or size of an object in space.

Why do we need to find a transformation matrix?

We need to find a transformation matrix in order to understand and manipulate geometric objects in a mathematical way. It allows us to perform transformations such as translation, rotation, and scaling on an object.

How do you find a transformation matrix?

To find a transformation matrix, you need to have two sets of coordinate systems - the original system and the transformed system. Then, you can use the coordinates of corresponding points in each system to construct the transformation matrix.

What is the purpose of a transformation matrix in linear algebra?

In linear algebra, a transformation matrix is used to represent and perform linear transformations on vectors and matrices. It helps us to understand and solve systems of equations, as well as perform operations such as rotation, reflection, and shearing.

Can a transformation matrix have different representations?

Yes, a transformation matrix can have different representations depending on the coordinate system being used. For example, a 2D transformation matrix will have a different representation than a 3D transformation matrix. Additionally, the order of operations in the transformation can also affect the representation of the matrix.

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