[Linear Algebra] $T$-invariant subspace

Please see attached image.
Advanced Linear Algebra exercise. Class is working out of Hoffman/Kunze Linear Algebra.
Any questions about notation or what theorems we have acess to, just let me know.

Some definitions:
(1) A linear operator $T$ over a vector space $V$ is diagonalizable if $V$ has a basis whose elements are eigenvectors of $T$. Alternatively, $T$ is diagonalizable if its eigenvectors generate $V$.
(2) Let $T$ denote a linear operator over a vector space $V$. A subspace $W$ of $V$ is $T$-invariant if $T(W) ? W$.
(3) If $W$ $\leq $ $V$ is  $T$-invariant (with $T$ $\in $ $L(V)$) the compression of $T$ to $W$ is $Tw:=$ $T|_{W}^{W} $ $\in $ $L(W)$.

Answer

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  • Hey, thank you very much. Two questions, by "base field" you mean F? What do you mean by rescaling w and z? Just making sure I got everything.

  • By base field I mean the field V is a vector space on; you didn't give it a name, but I guess F should do. By "rescaling w and z" I mean you can divide both w and z by a and get two vectors w' and z' such that w' = z' + e_i, and then "rename" w' and z' as w and z again.

  • I hope it is clear, if not just feel free to ask again!

  • Hey. A bit late, but I was revising this today and I have another question, if you have the time to break it down for me. Why exactly is $\{w_i : i\not\in J\}$ a basis for $W$?

  • If you'd prefer, I could go to your profile and ask a question directly.

  • Outside user interactions are prohibited. Alessandro Iraci will be notified automatically when you post a question.

  • Oh, sorry about that. I thought the ask a question button in a user's profile was asked exclusively to said user. Just finishing typing up the question. Would it be against the rules to post the answer Alessandro Iraci gave to ask for clarification on it?

  • No, it is fine.

  • ok, thank you!

  • Hi, I'm in the EU time zone, so I was sleeping when you posted your comment. I would have been happy to clarify, but since you asked another question it's more convenient if I answer that directly (so I can type math).

  • No worries, just pack 5 bucks worth of detail in there ;). I think I actually understood it a week ago, didn't even think to ask, but I'm drawing a blank revising now. Thank you!

  • I took it personally and wrote way more than what I think 5$ of details are worth, I hope it's fine now! :)

The answer is accepted.