linear algebra - if $T: V\to V$ and $ dim (KerT)+dim (ImT)=dimV $ can i ...
Mar 29, 2023 · But how can i proof that KerT ∩ ImT = {0} K e r T ∩ I m T = {0} or disprove it linear-algebra linear-transformations Share Cite
real analysis - Why doesn't IMT hold for all compact sets ...
Apr 8, 2020 · 0 In my college's notes, it says for all compact sets, extreme value theorem holds but intermediate value theorem doesn't. I wonder why since I think the original proof of IMT for f: [a, b] → …
Give an example of a linear map - Mathematics Stack Exchange
Jan 1, 2020 · Let's do an easy variation on your example, where V =R5 V = R 5 and W =R2 W = R 2. Define T T by T(x1,x2,x3,x4,x5) = (x1,x3) T (x 1, x 2, x 3, x 4, x 5) = (x 1, x 3).. Then clearly T T is …
Finding the basis of ker (T) and im (T) - Mathematics Stack Exchange
Jul 19, 2021 · 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 ...
Prove that $T^*$ is injective iff $ImT$ Is dense
Dec 21, 2014 · 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?
Is it true that if $T$ is a linear operator on a finite-dimensional ...
If you knew that $\ker T \cap \operatorname {im} T= \emptyset$, then you'd have a proof. But this isn't true, and you can easily find an example in small dimensional spaces.
Example of linear transformation on infinite dimensional vector space
May 22, 2018 · 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 V, and linear …
V = ImT \\oplus \\ KerT - Mathematics Stack Exchange
Linear Tranformation that preserves Direct sum V = ImT ⊕ KerT Ask Question Asked 12 years, 11 months ago Modified 12 years, 11 months ago
SageMath: Orthogonal projection of $\mathbb {C}^3$ onto a subspace.
Dec 13, 2024 · Now, my problem arises when I evaluate P_imT with specific values of a,b,c (in this case, the standard basis of $\mathbb {C}^3$) in order to obtain the columns of the projection matrix …
Limit of a complex function: Check all cases
Oct 28, 2024 · exists or not (where z0 z 0 is a complex number and f(z) f (z) is a complex function)", we generally travel along these paths (consider z0 = x0 +iy0 z 0 = x 0 + i y 0):: 1. 1. x→ x0 x → x 0 then …