Show all invariant subspaces are of the form

In summary, an invariant subspace is a subset of a vector space that remains unchanged under a specific transformation or operator. To show that all invariant subspaces are of the form, we can use the eigenvectors and eigenvalues of the operator to express them as a linear combination of basis vectors. This understanding of invariant subspaces has significance in understanding the structure of a vector space and making predictions about the behavior of the operator. Some common examples of invariant subspaces include the null space, column space, and row space of a matrix, as well as the space of stationary states in quantum mechanics. Invariant subspaces are an important concept in linear algebra as they allow for the breakdown of complex vector spaces into smaller, more manageable subspaces,
  • #1
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[solved] show all invariant subspaces are of the form

View attachment 1999

i don't even know how to begin (Angry)

C_x is a subspace spanned by x that belongs to V

C_x = {x, L(x), L^2(x),...}

edit: SOLVED
 

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  • #2
The hint refers to a question you asked in another thread. Do you remember it?
 
  • #3
dim of v needs to be equal to degree of minimal poly and hence that would be a contradiction?

i'll see what i can do with that. thanks.
 

FAQ: Show all invariant subspaces are of the form

1. What is an invariant subspace?

An invariant subspace is a subset of a vector space that remains unchanged under a particular transformation or operator. In other words, the elements of the subspace are mapped to themselves by the transformation.

2. How do you show that all invariant subspaces are of the form?

To show that all invariant subspaces are of the form, we must first prove that any subspace that is invariant under a particular operator can be expressed as a linear combination of a set of basis vectors. This can be done using the eigenvectors and eigenvalues of the operator.

3. What is the significance of showing that all invariant subspaces are of the form?

Showing that all invariant subspaces are of the form provides a way to understand the structure of a vector space and its relationship to the operator that acts on it. It also allows us to make predictions about the behavior of the operator and its effects on the subspace.

4. What are some common examples of invariant subspaces?

Some common examples of invariant subspaces include the null space, the column space, and the row space of a matrix. In quantum mechanics, the space of stationary states is also an invariant subspace under the time evolution operator.

5. How does the concept of invariant subspaces relate to linear algebra?

Invariant subspaces are an important concept in linear algebra as they allow us to break down a complex vector space into smaller, more manageable subspaces. This can help us to better understand the structure and behavior of a system and make predictions about its properties.

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