Determinant of a matrix and linear independence (explanation needed) It is written on Wikipedia that:
$n$ vectors in $\mathbb R^n$ are linearly independent if and only if the determinant of the matrix formed by taking the vectors as its columns is non-zero
Can someone explain this to me? You do not have to give a complete proof, just in simple terms explain what the determinant of that matrix has to do with linear independence? And why it has to be non-zero? And are vectors allowed to be rows instead of columns in that matrix?

Here a simple geometric explanation:


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*$2$ vectors in the plane are linearly independent if and only if they span a parallelogram with a non-zero area

*$3$ vectors in 3D-space are linearly independent if and only if they span a parallelepiped with a non-zero volume

*$n$ vectors in $\mathbb{R}^n$ are linearly independent if and only if they span an $n$-dimensional parallelepiped with a non-zero volume


The determinant is a so-called "volume form" that gives for $n$ vectors in $\mathbb{R}^n$ the $n$-dimensional volume of the parallelepiped that is spanned by those vectors (up to a sign which gives rise to the so called orientation). 
So, if $n$ vectors in $\mathbb{R}^n$ are linearly dependent, they cannot span an $n$-dimensional parallelepiped and hence produce a volume of zero.
