Let $x_{1},......,x_{n}$ be a set of $n$ vectors over a real or complex field $F$ in $V.$ Consider the matrix
$$ \begin{pmatrix} \langle x_{1},x_{1} \rangle & \langle x_{2},x_{1} \rangle &.........&\langle x_{n},x_{1} \rangle \\ \langle x_{1},x_{2} \rangle & \langle x_{2},x_{2} \rangle &.........&\langle x_{n},x_{2} \rangle \\ &.......................&\\ \langle x_{1},x_{n} \rangle & \langle x_{2},x_{n} \rangle &.........&\langle x_{n},x_{n} \rangle \\ \end{pmatrix}.$$ Then
$x_{1},......,x_{n}$ are Linearly independent iff $\det A=0$
$x_{1},......,x_{n}$ are Linearly independent iff $\det A>0$
$\det A \geq0,$ always
$\det A\leq0,$ always
I think option 2 and 3 is correct but how can i approach this kind of question. thanks in advance.
