Let $(V,\langle,\rangle)$ be a finite dimensional vector space. Let $T$ be a linear operator on V. Show that $\operatorname{Im}T=(\ker T^*)^\perp$
45998
The range of $T^*$ is the orthogonal complement of $\ker(T)$
92291
$U_1, U_2, ..., U_k$ subspaces of linear space $X$ with same dimensions, prove $X=U_1\oplus V=U_2\oplus V=...=U_k\oplus V$
56211
An operator that commutes with another operator $T$ with distinct characteristic values is a polynomial in $T$
20879
Let $T:V→V$ be a linear transformation satisfying $T^2(v)=-v$ for all $v\in V$. How can we show that $n$ is even?
112024
Let $T:V → V$ be a linear transformation where $V$ is finite dimensional.
144104
Show that a positive operator on a complex Hilbert space is self-adjoint
46265
Minimal polynomial of the operator $T:V\oplus W\to V\oplus W$
103839
Proving that $T^{2} = T$ for a linear operator on $W$ implies that $V = \operatorname{Null} T \oplus \operatorname{Range} T$
90757
$T$ self-adjoint on Hilbert space $H$, $T^n=T^{n+1}$ will $T$ be an orthogonal projection operator?
34147
$\|u\|\leq \|u+av\| \Longrightarrow \langle u,v\rangle=0$
27901
Proving $\sum\limits_{i=1}^k | \langle x,v_i \rangle \langle y,v_i\rangle| \leq \|x\|\cdot \|y\|$
80346
If every eigenvector of $T$ is also an eigenvector of $T^{*}$ then $T$ is a normal operator
90475
Coordinate-free proof of $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$?
68449
$\operatorname{Supp}(M)=V(\operatorname{Ann}M)$ if $M$ is finitely generated
156858
Kernel of powers of a normal operator
97158
Prove that the operator norm is a norm
31919
What can we say about the inner product $\langle f(t),f'(t) \rangle$?