Proving the Relationship Between Determinants and Hyperplanes

[Harmony]

Homework Statement

[PLAIN]http://img835.imageshack.us/img835/1108/question.png

The Attempt at a Solution

I understand that the determinant represents the set of points, x in R_k which lies in the hyper plane passing through the points p1, p2...p_k. I also know, that if the determinant is non zero, and the sign is opposite for two choice of x, then they must be on different side of the hyper plane.

However, i am not too sure how can i prove this. My first thought is that this looks like the determinant to a Jacobian, but can't proceed much beyond that.

Any hints and helps is appreciated. Thanks.

 

[HallsofIvy]

Since the problem asks specifically about the cases k= 2 and k= 3 did you look at them? Often when a problem asks about "2" and "3" and then "n", the first two are intended as a hint.

With k= 2, this equation becomes
$$\left|\begin{array}{ccc}1 & 1 & 1 \\ x & p_{1x} & p_{2x} \\ y & p_{1y} & p_{2y}\end{array}\right|= \left|\begin{array}{cc}p_{1x} & p_{2x} \\ p_{1y} & p_{2y}\end{array}\right|- \left|\begin{array}{cc}x & p_{2x} \\ y & p_{2y} \end{array}\right|+ \left|\begin{array}{cc}x & p_{1x} \\ y & p_{1y}\end{array}\right|$$
where I have expanded by the first row.
You can continue but it should be obvious that you will have a linear equation in x and y- the equation of a line.

Similarly, in three dimensions, expanding by the first row gives you numbers multiplying in x, y, and z, again a linear equation. That will be the equation of a plane.

Since there is only one "x" in the determinant, no matter what k is that reduces to a linear equation in the k components of x so gives a k-1 dimensional linear manifold, a "hyperplane" as you titled this thread. (But not necessarily a subspace. The set will contain the origin only if the determinant of the given "p"s is 0.)

 

[Harmony]

[PLAIN]http://img823.imageshack.us/img823/1883/questiona.png

Thanks for your help. I am stuck with the next part of the question though. I know that a and b will lies on the different side of the line/plane/hyper plane, but I am not too sure how to provide a rigorous proof for that...