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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?

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  • $\begingroup$ If the volume is zero, the vector taht define the (flat) parallelepiped are not linearly independent. $\endgroup$ – Bernard Jan 4 '15 at 16:27
  • $\begingroup$ @Bernard: But that is what I wrote: They are linearly dependent?!? $\endgroup$ – vonjd Jan 4 '15 at 16:29
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The volume of the parallelopiped so formed is zero(ie., determinant is zero) means that the lines represented by the linear equations will be on the same plane.

Observe that if there are two lines in a plane means it must intersect or parallel.

Here all are homogeneous equations means each one is a subspace of $\mathbb R^k$. So, each line has zero as a point. Hence, these lines are intersecting lines. If two lines are intersecting more than one point, then it must intersect with all other points which gives the determinant is zero.

In this case, obviously the volume of the parallelopiped is zero, since the two lines are coinsides.

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