Linear Transformations proof

In summary, the conversation discusses finding the eigenvalues and eigenvectors for a linear transformation on a vector space of functions, where the transformation is defined by taking derivatives. The first part focuses on finding the eigenspace for real eigenvalues, while the second part proves that all real numbers are eigenvalues for a specific transformation. The conversation also mentions using differential equations to solve for the functions that satisfy the transformation.
  • #1
kehler
104
0

Homework Statement


Let V be the vector space of all functions f: R->R which can be differentiated arbitrarily many times.
a)Let T:V->V be the linear transformation defined by T(f) = f'. Find the (real) eigenvalues and eigenvectors of T. More precisely, for each real eigenvalue describe the eigenspace of T corresponding to the eigenvalue. (Since the elements of the vector space are functions, some people would use the term eigenfunction instead of eigenvector
b)Now let T:V->V be the linear transformation defined by T(f) = f''. Prove that every real number eigenvalue is an eigenvalue of T


The Attempt at a Solution


I don't really know where to start :S
The eigenvectors should be functions f where
T(f) = eigenvalue x f
So, f' = eigenvalue x f
What do I do from here? I can't expand it cos I don't know the form of f(x)
 
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  • #2
Well I can think of a class of functions for which f(n) = [tex]\lambda[/tex]f. Though I don't know if they are the only possible types of functions which satisfy this property.
 
  • #3
Would that be the exponential functions?
 
  • #4
In other words, you need to solve the differential equations [itex]df/dx= \lambda f[/itex] and [itex]d^f/dx^2= \lambda f[/itex]. That shouldn't be too hard.
 

1. What is a linear transformation?

A linear transformation is a mathematical function that maps vectors from one vector space to another. It preserves the operations of vector addition and scalar multiplication, meaning that the linear combination of two vectors is equal to the transformation of the linear combination of those two vectors.

2. How do you prove that a transformation is linear?

To prove that a transformation is linear, you must show that it satisfies two conditions: 1) f(u+v) = f(u) + f(v) for all vectors u and v, and 2) f(cu) = cf(u) for all vectors u and scalar c. This means that the transformation must preserve vector addition and scalar multiplication.

3. What is the significance of the standard matrix in linear transformations?

The standard matrix of a linear transformation is a matrix representation of the transformation. It allows us to easily perform calculations and transformations on vectors without having to use the original function. It also helps us to visualize and understand the effects of the transformation on vectors.

4. Can a linear transformation be represented by a non-square matrix?

Yes, a linear transformation can be represented by a non-square matrix. The size of the matrix depends on the dimension of the vector spaces involved in the transformation. For example, a transformation from R^3 to R^2 can be represented by a 2x3 matrix.

5. How are composition of linear transformations and matrix multiplication related?

The composition of two linear transformations is equivalent to the matrix multiplication of their standard matrices. This means that if we have two linear transformations, T1 and T2, with standard matrices A and B respectively, the composition T2(T1(x)) is equal to the matrix multiplication BAx. This relationship allows us to easily perform compositions of linear transformations using matrix multiplication.

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