Show $φ^n= 0$ for $φ:V→V$ a linear endomorphism such that $\mathrm{tr}(\wedge^qφ) = 0$ and $\dim(V) = n$ a vector space.
1973
Traces of all positive powers of a matrix are zero implies it is nilpotent
97608
Showing a Ring of endomorphisms is isomorphic to a Ring
141166
How to show that when the linear transformation $\Phi =\psi \circ \phi : E \rightarrow G$ is injective, then $\phi$ is also injective?
61013
Diffeomorphism between the Grassmannian manifolds $\mathbf{Gr}(n,k)$ and $\mathbf{Gr}(n,n-k)$.
154142
Finding a specific basis for an endomorphism
65162
Show that $\dim H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m)) = {n + m \choose n}$ if $m \geq 0$, and $0$ otherwise.
54737
Proof that $V = \sum_k U_k$ and $\dim V=\sum_k \dim U_k$ implies $V=\oplus_k U_k$
25009
$\operatorname{tr}(A)=\operatorname{tr}(A^{2})= \ldots = \operatorname{tr}(A^{n})=0$ implies $A$ is nilpotent
152887
Let $T : V \rightarrow V$ be a linear operator. If $T^n = O_V$ for some $n \ge 1$, prove that $I_V + T$ is an isomorphism.
42334
Does $\lambda$ $\mathbf u$ $= 0$ imply $\lambda = 0$ or $\mathbf u$ $= 0$?
62950
Use implicit function theorem to show $O(n)$ is a manifold
20467
Show that $(y_n)\in l^q$
22149
How to prove $dim(Ker (A- \lambda I))=dim (Ker(A^t- \lambda I))$?
127000
When does $V = \ker(f) \oplus \operatorname{im}(f)$?
98436
Finite order endomorphisms
103308
Showing that $\nabla\times(\nabla\times\vec{A}) = \nabla(\nabla\cdot\vec{A})-\Delta\vec{A}$
4652
Is $V$ a simple $\text{End}_kV$-module?