For $u,v$ nonzero, show $u\cdot v = 0$ implies $u$ and $v$ are linearly independent.
30720
If the dot product between two vectors is $0$, are the two linearly independent?
150141
Showing $X$, $Y$ are not independent but $\operatorname{Cov}[X, Y] = 0$
99105
Are the functions $\sin^n(x)$ linearly independent?
24900
Homogeneity and Differentiability at $0$ implies linearity?
121558
Showing that $n$ exponential functions are linearly independent.
112231
Showing that a one-to-one linear transformation maps a linearly independent set onto a linearly independent set
119947
Can a non-zero vector have zero image under every linear functional?
56102
Prove that there is no non-zero linear operator on $\mathbb C^{2}$ such that $(\alpha | T\alpha) = 0$
58774
There exist an infinite subset $S\subseteq\mathbb{R}^3$ such that any three vectors in $S$ are linearly independent.
127052
Independence of non-constant real variables
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.
97049
If $v_1,...,v_m$ are linearly independent, then the span $v_1+w,...,v_m+w$ has dimension $\ge m-1$
98913
Sign in $\Delta U= -\int\vec{F}\cdot d\vec{l}$
42334
Does $\lambda$ $\mathbf u$ $= 0$ imply $\lambda = 0$ or $\mathbf u$ $= 0$?
150720
Given that $W$ is a subspace of $V$, prove: $\dim W=\dim V \implies W=V$
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($\mathbb{R},\mathbb{R}$)