I understand the solution of a linear system of equation is equivalent to finding the intersection points of n-hyperplanes.

There are 3 elementary row operations - scaling an equation, exchanging equations and subtracting a scalar multiple of an equation from another equation.

The first two I understand geometrically. I am trying to understand the subtraction of two equations geometrically. I can see that the row operation produces a new hyperplane which has one of the coordinate axes as its normal. What I am specifically trying to understand is how the hyperplane is rotated by exactly the right angle?. Anything to do with direction cosines/

Thanks Rajagopal


The coefficients of the equation specify the normal vector of the corresponding hyperplane. Thus, if you add equations, you're adding their normal vectors. Specifically, if you have equations $e_1$ and $e_2$ and you form $e=e_1+\lambda e_2$, you get a new normal vector $n=n_1+\lambda n_2$. The direction of this new normal vector goes to that of $n_2$ for $\lambda\to\pm\infty$, but, unless $n_2$ is parallel to $n_1$, it is never that of $n_2$, which ensures that the new equation is linearly independent of the one for $n_2$ if the old one was.

By "the row operation produces a new hyperplane which has one of the coordinate axes as its normal", I presume you're referring specifically to the row operations used in Gaussian elimination which make one of the coefficients $0$. This doesn't mean that the normal of the new hyperplane is one of the coordinate axes (else it would have to have all the previous coordinate axes as its normal, too). Rather, it means that the normal of the hyperplane is orthogonal to that coordinate axis.

  • $\begingroup$ thanks, thinking about it in terms of the normal helps. as an aside, what i was referring to is the following If we have 3x - y = 1 and x + 2y = 5 We can then do 3*eqn.2 - eqn 1 to get 7Y = 14. This is a plane that has the y-axis has its normal. $\endgroup$
    – Rajagopal
    May 23 '11 at 8:39
  • $\begingroup$ @Rajagopal: That only happens in the very last step of the elimination. In all the previous steps, all you can say when you produce a $0$ by subtraction is that you've made the normal vector orthogonal to one of the axes. In the last step, when you've made it orthogonal to all the axes except one, it is then necessarily parallel to that remaining axis. $\endgroup$
    – joriki
    May 23 '11 at 8:52

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