Linear Algebra Linear Algebra Math Forum

 May 11th, 2018, 01:30 AM #1 Newbie   Joined: Mar 2018 From: Split, Croatia Posts: 13 Thanks: 0 Sum of two invariant subspaces Let f : V->V be linear mapping and L,M f-invariant subspaces of V. Prove that L+M is also f-invariant. I know how things work for special case when L is the direct complement of M, but I'm stuck with this L+M. May 11th, 2018, 07:16 AM #2 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125 Suppose that T:V→V is a linear transformation with eigenvalue λ and associated eigenspace ET(λ). Let W be any subspace of ET(λ). Then W is an invariant subspace of V relative to T. http://linear.ups.edu/version3/scla/section-IS.html Thanks from Birgitta May 11th, 2018, 10:02 PM   #3
Senior Member

Joined: Sep 2016
From: USA

Posts: 624
Thanks: 396

Math Focus: Dynamical systems, analytic function theory, numerics
Quote:
 Originally Posted by zylo Suppose that T:V→V is a linear transformation with eigenvalue λ and associated eigenspace ET(λ). Let W be any subspace of ET(λ). Then W is an invariant subspace of V relative to T. http://linear.ups.edu/version3/scla/section-IS.html
Not every invariant subspace is spanned by eigenvectors.

The exercise is essentially one line. Can you explain how you are showing this for the case that $L \oplus M = V$? I think you should find that you never need that assumption. May 13th, 2018, 06:19 AM   #4
Newbie

Joined: Mar 2018
From: Split, Croatia

Posts: 13
Thanks: 0

I think I got the idea

I finally proved this, hope it is ok? Thank you
Attached Images 20180513_161554.jpg (99.7 KB, 7 views) May 13th, 2018, 08:47 AM #5 Senior Member   Joined: Sep 2016 From: USA Posts: 624 Thanks: 396 Math Focus: Dynamical systems, analytic function theory, numerics yep you got it. Thanks from Birgitta May 14th, 2018, 08:27 AM   #6
Banned Camp

Joined: Mar 2015
From: New Jersey

Posts: 1,720
Thanks: 125

Quote:
 Originally Posted by Birgitta Let f : V->V be linear mapping and L,M f-invariant subspaces of V. Prove that L+M is also f-invariant.
I acknowledge Brigitta's correct answer. However, I think it should start with the definition of invariant subspace.

Definition: A subspace W of a vector space V is said to be invariant with respect to a linear
transformation T ∈ L(V, V ) if T (W) ⊆ W. *

The definition virtually implies the answer:
If x ∈ U and y ∈ W,
T(ax+by)=aTx+bTy belongs to U+W because aTx belongs to U and bTy belongs to W, by definition.

And U+W is a subspace, because if ax1=ay1+az1 and bx2 =by2+bz2 belong to U+W, so does ax1+bx2=a(y1+y2)+b(z1+z2).

* https://math.okstate.edu/people/bine...3-5023-l18.pdf Tags invariant, subspaces, sum Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post raul21 Math 1 May 28th, 2014 03:03 AM HubertM Real Analysis 1 January 19th, 2014 01:19 PM hossein-pnt Real Analysis 3 December 21st, 2012 10:28 AM mami Linear Algebra 0 April 27th, 2012 02:11 PM tinynerdi Linear Algebra 0 April 11th, 2010 10:48 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top       