What is the geometric interpretation of the determinant of a matrix representing a system of homogenous linear equations?
We know that
- iff the determinant is equal to zero the system has a non-trivial solution and
- in this case the volume of the parallelepiped determined by the column/row vectors of the matrix is also zero (which means that these are linearly dependent).
How can one see the connection between these two facts? How does a volume being zero lead to a non-trivial solution?