If $v$ and $w$ are orthogonal to each other with respect to the scalar product $\langle \cdot, \cdot \rangle$, they are also linearly independent.
30017
Orthogonality and linear independence
121853
$X$,$Y$,$Z$ mutually independent implies $X+Y$ independent of $Z$
131998
let ${v_1}, {v_2} $ be orthonormal basis for subspace $W$. Prove that if $z$ is a vector in $W$, then
71316
Two inner products being equal up to a scalar
71064
How to prove that $\langle Av,w\rangle=\langle v,A^Tw\rangle$?
152932
Proof: $\det\pmatrix{\langle v_i , v_j \rangle}\neq0$ $\iff \{v_1,\dots,v_n\}~\text{l.i.}$
112284
If $\mid a_{jj}\mid \gt \sum_{i \neq j} \mid a_{ij} \mid$ then vectors $a_1,\dots ,a_n \in \mathbb{R}^n$ are linearly indendent.
62077
Prove that if $W_1\subseteq V$ finite-dimensional, then there is $W_2\subseteq V$ such that $V=W_1\oplus W_2$
92600
Let $V$ be a $K$-vector space, $f: V \rightarrow V$ a linear map. Under some condition, show that $v, f(v),...,f^n(v)$ are linearly independent.
37794
Proving the two given vectors are orthogonal
50159
How to prove if $\Bbb R^n=W_1\cup W_2\cup \cdots\cup $ where each $W_k$ is a subspace, then $\Bbb R^n=W_i$ holds for some $i$
78997
If $L/K$ normal and $H = \operatorname{Aut}(L/K)$, then $L/L^H$ is separable and $L^H/K$ is purely inseparable.
138343
How to prove that the set $\{\sin(x),\sin(2x),...,\sin(mx)\}$ is linearly independent?
96354
Linear independence of $\sin(x)$, $\sin(2x)$, $\sin(3x)$ in Map($\mathbb{R},\mathbb{R}$)
24585
Proving that $\mathbf{W}$+$\mathbf{W^{\perp}}$=$\mathbb{R^{n}}$
97994
Orthogonality Relations Exercise, Brezis' Book Functional Analysis
135641
Algebraic independence of exponentials