Let $Q$ be orthogonal matrix and $v,u$ be vectors. Show that $Qu \cdot Qv = u \cdot v$
141015
Proof of orthogonal matrix
30877
How to show that the set of orthogonal n x n matrices forms a group under multiplication
49296
Prove $|(1+|u|^2)v-(1+|v|^2)u|>|u\bar{v}-\bar{u}v|$
131609
Prove that $Au\cdot v = u\cdot A^Tv$
155082
If $P$ is real orthogonal matrix with $\det P = -1$, prove that $-1$ is an eigenvalue of $P$.
152932
Proof: $\det\pmatrix{\langle v_i , v_j \rangle}\neq0$ $\iff \{v_1,\dots,v_n\}~\text{l.i.}$
54737
Proof that $V = \sum_k U_k$ and $\dim V=\sum_k \dim U_k$ implies $V=\oplus_k U_k$
118049
Show that $\mathbb{R}^{\mathbb{R}} = U_{e} \oplus U_{o}$
90345
If $\lVert x\rVert=\lVert y \rVert$, prove that exists a unitary transformation such as $Tx=y$
132386
Show that $(\mathbb{Q}^*,\cdot)$ and $(\mathbb{R}^*,\cdot)$ aren't cyclic
133087
Prove that, if $\{u,v,w\}$ is a basis for a vector space $V$, then so is $\{u+v, v+w, u+v+w\}$.
1865
Sub-determinants of an orthogonal matrix
59598
The vectors are orthonormal
117175
Proof that the 2-norm of orthogonal transformation of a matrix is invariant
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 $T^n$
62065
Prove that $v_1, \dots v_n$ is a basis of V.
9003
prove $\frac{1}{2}\mathbf{\nabla (u \cdot u) = u \times (\nabla \times u ) + (u \cdot \nabla)u}$ using index notation