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4.1 Area, Volume and the Determinant in Two and Three Dimensions

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In two dimensional space there is a simple formula for the area of a parallelogram bounded by vectors v and w with v = (a, b) and w = (c, d): namely ad - bc
Why?
1. The statement that the area of a parallelogram with sides given by the vectors (a, b) and (c, d) is |ad - bc| is obviously true if b and c are 0, since the parallelogram is then a rectangle with sides |a| and |d|, whose area is |ad|.
2. Neither the area nor ad - bc changes if we add a multiple of (a, b) to (c, d) or vice versa. The parallelogram merely tilts, its base and altitude remain the same. If we for example, add a to c, we get a change in ad - bc of -ba; but adding b to d produces a compensating change of ab to it, and the net effect is 0.
3. By repeatedly adding such multiples we can force b and c to be 0, after which the statement holds.
4. Since these addings didn't change anything, it must have been true at the beginning.
The combination ad - bc is called the determinant of the matrix:

Given three vectors in three dimensions we can form a 3 by 3 matrix of their components, and the absolute value of the determinant of that matrix will be the volume of the parallelepiped whose edges are determined by the three vectors. In fact an analogous results holds for k k-vectors  and determinants and the k-volume of the figure they bound.
What then is the determinant of a matrix?
In any dimension, it is defined as follows:
1. it is linear in the elements of any row (or column)  so that multiplying everything in that row by z multiplies the determinant by z, and the determinant with row v + w  is the sum of the determinants otherwise identical with that row being v and that row being w.
2. It changes sign if two of its rows are interchanged (it must therefore be 0 if two rows are identical).
3. The matrix with 1’s on the diagonal and 0’s elsewhere has determinant 1.

The determinant is generally written as det M or as | M | or sometimes || M ||.
If v and w are two rows of the matrix M we can deduce from the first two conditions that adding a multiple of v to w does not change the determinant of M.
The volume of a parallelepiped bounded by edges whose directions and lengths are that of u, v and w is almost linear in u, v and w; it differs from linearity only in that it is always positive, like length in one dimension is. It is 1 if the vectors have unit length and are mutually perpendicular, and does not change if one side is added to another; The absolute value of the determinant of the matrix formed by the components of the three vectors obeys the same conditions and is therefore the same thing.
In higher dimensions the analog of volume is called hypervolume and the same conclusion can be drawn by the same argument: The hypervolume of the parallel sided region determined by k vectors in k dimensions is the absolute value of the determinant whose entries are their components in the directions of the (orthonormal) basis vectors.
In fact, the determinant can be considered a linear and signed version of hypervolume.
Consider the hypervolume of a parallel-sided region with sides xA,B,C,... as a function of x. It is linear in x for positive or negative x, but it is always positive, and its graph looks like a V, taking the value 0 for x = 0.
The determinant is the same as the hypervolume for positive or negative x and minus the hypervolume for the other, and is therefore linear as a function of x.