A problem in Hoffman's Linear Algebra

In summary, the problem in Hoffman's Linear Algebra on Page 243 involves the relationship between diagonalizable linear operators and T-invariant subspaces. It is stated that if T is diagonalizable, then every T-invariant subspace has a complementary T-invariant subspace, and vice versa. To prove this, one can use the concept of T-admissibility or utilize the fact that T is "self adjoint" and show that any T-invariant subspace is orthogonal to its complementary subspace. The question then arises, what is the inverse of this proposition?
  • #1
tghg
13
0
A problem in Hoffman's Linear Algebra.
Page 243

18. If T is a diagonalizable linear operator, then every T-invariant subspace has a complementary T-invariant subspace. And vice versa.

In fact, the answer lies on Pages 263~265,but I try not to use the conception T-admissible to prove this proposition.
Could someone help me out?
 
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  • #2
There are a couple of different ways to do that:

The fact that T is "diagonalizable" means that there exist a basis for the vector space consisting entirely of eigenvectors of T (so that the matrix for T in that basis is diagonal). Using that, clearly any T-invariant subspace is spanned by some subset of those eigenvectors and it's orthogonal complement is spanned by the remaining eigenvectors- and so is T-invariant itself.

Or you could use the fact that, since T is diagonalizable, it is "self adjoint": for any vector u,v <Tu, v>= <u, Tv> where <u, v> is the inner product of u and v. That should make it easy to show that if a subspaced is T-invariant, then so is its orthogonal complement.
 
  • #3
How about the Inversion of the proposition?
 

Related to A problem in Hoffman's Linear Algebra

1. What is the nature of Hoffman's Linear Algebra problem?

Hoffman's Linear Algebra problem is a mathematical concept that deals with the study of linear equations and their properties, such as solutions, matrices, and vector spaces. It involves solving systems of linear equations and representing them using matrices and vectors.

2. What makes Hoffman's Linear Algebra problem challenging?

Hoffman's Linear Algebra problem can be challenging due to its complex nature and the use of abstract mathematical concepts. It requires a deep understanding of linear algebra, as well as critical thinking and problem-solving skills.

3. How is Hoffman's Linear Algebra problem applied in real life?

Hoffman's Linear Algebra problem has many practical applications in fields such as engineering, computer science, economics, and physics. It is used to model and solve real-world problems involving systems of linear equations, such as optimizing resources and predicting trends.

4. What are some common techniques used to solve Hoffman's Linear Algebra problem?

Some common techniques used to solve Hoffman's Linear Algebra problem include Gaussian elimination, matrix operations, and vector spaces. These techniques involve manipulating equations and matrices to find solutions to systems of linear equations.

5. Are there any helpful resources for learning more about Hoffman's Linear Algebra problem?

Yes, there are many helpful resources available for learning more about Hoffman's Linear Algebra problem, such as textbooks, online courses, and tutorials. It is also beneficial to practice solving problems and seeking help from a math tutor or professor if needed.

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