Linear dependence (Homework 1, Physics 325)

Are the following sets of vectors given by there coordinates in some bases linearly dependent or

independent?

a) (2,3,0), (0,0,1),(2i, i,-i)

b) (0,4,0), (i,3i, i), (2,0,1)

c) (i,1,2),(3, i,1), (i,3i,5i)

$\begin{vmatrix}

2 & 0 & 2i\\

-3 & 0 & i\\

0 & 1 & -i

\end{vmatrix}=0-6i+0-0-2i-3i=-11i\neq 0$

so vectors are linearly independent.

$\begin{vmatrix}

0 & i& 2\\

4 & 3i & 0\\

0 & i &1

\end{vmatrix}=0-8i+0+6i-0-4i=-10i\neq 0$

so vectors are linearly independent.

$\begin{vmatrix}

i & 3 & i\\

1 & i & 3i\\

2 & 1 &5i

\end{vmatrix}=-5i+i+18i-2-3-15i=-5-i\neq 0$

so vectors are linearly independent.

If $|\psi>=|\phi_1>+|\phi_2>$ and $|\chi>=|\phi_1>-|\phi_2>$, where $|\phi_1>$, $|\phi_2>$ ARE NOT ORTHOGONAL, prove the following relations

a)$<\psi|\psi>+<\chi|\chi>=2<\phi_1|\phi_1>+2<\phi_2|\phi_2>$

b) $<\psi|\psi>-<\chi|\chi>=2<\phi_1|\phi_2>+2<\phi_2|\phi_1>$

$|ψ>=|ϕ_1>+|ϕ_2>$     and $|χ>=|ϕ_1>-|ϕ_2>$
$<ψ|ψ> = <ϕ_1 | ϕ_1>+<ϕ_1 |ϕ_2>+<ϕ_2 | ϕ_1>+<ϕ_2 |ϕ_2>$

$<χ|χ> = <ϕ_1 | ϕ_1>-<ϕ_1 |ϕ_2>-<ϕ_2 | ϕ_1>+<ϕ_2 |ϕ_2>$

So that

$<ψ|ψ> +<χ|χ> =2<ϕ_1 |ϕ_1> +2<ϕ_2 | ϕ_2>$

$<ψ|ψ> -<χ|χ> =2<ϕ_1 |ϕ_2> +2<ϕ_2 | ϕ_1>$