$n=\dim V$. Then $V=\ker(T^n)\oplus\mathrm{range}(T^n)$
62303
For $V$ vector space of dimension $n$ under $\mathbb C$ and $T: V \to V $ linear transformation , show $V= \ker T^n \oplus $ Im $T^n$
132492
If a linear transformation $T$ has $z^n$ as the minimal polynomial, there is a vector $v$ such that $v, Tv,\dots, T^{n-1}v$ are linearly independent
87494
$\operatorname{rank}(T^3) + \operatorname{rank}(T) \geq 2\operatorname{rank}(T^2)$ for finite-dimensional $V$
109228
Prove that $v, Tv, T^2v, ... , T^{m-1}v$ is linearly independent
92291
$U_1, U_2, ..., U_k$ subspaces of linear space $X$ with same dimensions, prove $X=U_1\oplus V=U_2\oplus V=...=U_k\oplus V$
62065
Prove that $v_1, \dots v_n$ is a basis of V.
55488
Proving that $\operatorname{codim}(X,Y)=\dim(\mathcal{O}_{X,\eta})$
131998
let ${v_1}, {v_2} $ be orthonormal basis for subspace $W$. Prove that if $z$ is a vector in $W$, then
46265
Minimal polynomial of the operator $T:V\oplus W\to V\oplus W$
146541
Why $\dim U+\dim U^\perp=\dim V$?
114575
If $\sum_{i=1}^k \dim F_i > n(k-1)$ then $\cap_{i=1}^k F_i \neq \{0\}$
153406
$[F(t):F(t^n)]=n$ where $t$ is trascendental
99300
Prove if $W \subset V$ is a subspace such that $T(W)\subset W$, then $V/W$ is finite-dimensional.
48291
Proof that $\mathrm{ker}(f^*) = \mathrm{im}(f)^{\perp} $
15909
Bijection from $\mathbb R$ to $\mathbb {R^N}$
95379
Prove that $V = V_1 \oplus V_2$ in the following
34520
Is $0\,\mathrm{m}$ dimensionless?