Linear Independence

Let “linearly independent states” be states that you cannot create another linearly independent state from multiplying and adding another linearly independent state.

a cleaner definition is ∣ψ1⟩,∣ψ2⟩,…∣ψn⟩\left\lvert \psi_1 \right\rangle,\left\lvert \psi_2 \right\rangle,…\left\lvert \psi_n \right\rangle are linearly independent if the only way to get

c1∣ψ1⟩+c2∣ψ2⟩+⋯+cn∣ψn⟩=0\begin{gather*} c_1\left\lvert \psi_1 \right\rangle+c_2\left\lvert \psi_2 \right\rangle+\cdots+c_n\left\lvert \psi_n \right\rangle=0 \end{gather*}

is having ci=0c_i=0 for all (∀\forall) cic_i.

Example:

Given c∈Rc \in \mathbb{R} Standard Number System

∣+⟩=c∣−⟩will never happen how much you try\begin{gather*} \left\lvert + \right\rangle=c\left\lvert - \right\rangle\quad\text{will never happen how much you try} \end{gather*}

and vice versa.