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- linear algebra - if $T: V\to V$ and $ dim (KerT)+dim (ImT)=dimV $ can i . . .
$KerT+ImT=dimV$ ? Is this possible? $Ker T, Im T$ are subspaces of $V$ and $dimV$ is a just a
- V = ImT \\oplus \\ KerT - Mathematics Stack Exchange
Linear Tranformation that preserves Direct sum V = ImT ⊕ KerT Ask Question Asked 12 years, 10 months ago Modified 12 years, 10 months ago
- Prove that $T^*$ is injective iff $ImT$ Is dense
The title of your question does not really match the actual question (maybe the statement of the current question is used to prove the result in the title?) Is this intended?
- linear algebra - Prove Ker$T= ($Im$T^*)^\bot$ and (Ker$T^*$)$^\bot . . .
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges,
- Finding the basis of ker (T) and im (T) - Mathematics Stack Exchange
for part d, would elaborate by showing that the image of $T$ is equal to the span of $\ {1,x\}$ Since you already know that $1$ and $x$ are linearly independent
- Find a basis for KerT and ImT (T is a linear transformation)
Find a basis for KerT and ImT (T is a linear transformation) Ask Question Asked 6 years, 5 months ago Modified 6 years, 5 months ago
- Give an example of a linear map $T$ such that $\dim (\operatorname . . .
This is completely correct This will give a linear map with the properties you're asked for I think that it is a bit too general to actually be "an example" I think it would be better if you actually pick a concrete basis But that's a personal aesthetic belief, and one would have to be pretty pedantic about it to say that that makes you wrong One objection with a bit more substance is
- Example of linear transformation on infinite dimensional vector space
I haven't had much experience with infinite dimensional vector spaces, and I was working on a problem that asks to prove that for a finite dimensional vector space $V$, and linear transformation $T:V\to V$, $V=imT + ker T \implies V=imT \bigoplus ker T$
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