Does $\dim V=\dim(\operatorname{Ker}(T))+\dim(\operatorname{Im}(T))$ imply $V=\operatorname{Ker}(T)+\operatorname{Im}(T)$ iff $T$ is a homomorphism?
127000
When does $V = \ker(f) \oplus \operatorname{im}(f)$?
118354
Epimorphism and Monomorphism = Isomorphism?
52354
Given a idempotent endomorphism, show that $M = \ker(f) \oplus \ker (f-\text{id}_M)$
74675
Why is there no natural isomorphism between $V$ and its dual?
123419
Abstract algebra: homomorphism and an element of a prime order
45179
When does the isomorphism $G\simeq \ker(\phi)\times \operatorname{im}(\phi)$? hold?
42810
Is $(\mathbb{Q},+)$ isomorphic to $(\mathbb{Q}^n,+)$?
126644
Simple module and homomorphisms
85239
Proof $dim(W)=dim(V)=n \implies W=V$
50130
$\operatorname{Aut}(V)$ is isomorphic to $S_3$
7890
Prove the map $\Phi:G\to\operatorname{Aut}G$ with $g\mapsto (x\mapsto g^{-1}xg)$ is an homomorphism.
117131
Canonical Isomorphism Between $\mathbf{V}$ and $(\mathbf{V}^*)^*$
53633
Given two subspaces $U,W$ of vector space $V$, how to show that $\dim(U)+\dim(W)=\dim(U+W)+\dim(U\cap W)$
34520
Is $0\,\mathrm{m}$ dimensionless?
89177
$\operatorname{adj}(AB) = \operatorname{adj} B \operatorname{adj} A$
91222
Isomorphism between $\mathfrak o(4,\mathbb R)$ and $\mathfrak o (3,\mathbb R) \oplus\mathfrak o (3,\mathbb R) $
1026
Ring homomorphism with $\phi(1_R) \neq1_S$