Re: LinAlg: Vector Spaces - need help understanding an example

From: Henrik Bergstr?m (henrikb_at_shell.linux.se)
Date: 08/16/04 

Date: 16 Aug 2004 04:53:29 -0700

Virgil <ITSnetNOTcom#virgil@COMCAST.com> wrote in message news:<ITSnetNOTcom#virgil-35732F.14010312082004@comcast.dca.giganews.com>...
> In article <141d398b.0408120731.7c679670@posting.google.com>,
> henrikb@shell.linux.se (Henrik Bergstr?m) wrote:
>
> > Yes, it's the same as the original system, but with the constant of
> > each equation set to 0, i.e. one "sets the right side to zeros". So, I
> > know what a homogenous system is but it seems as if I still haven't
> > fully grasped all its implications.
>
> Consider the single homogeneous equation x + y = 0
> versus the single non-homogeneous equation x + y = 1.
>
> For the first (homogeneous) equation, x + y = 0,
> x = 1, y = -1 and x = 3, y = 3 are solutions,
> and x = 1 + 3 = 4, y = -1 + -3 = -4 is also a solution, so "adding"
> those solutions gives another solution. This always happens.

This is interesting and I see that this is the case by "doing the
algebra" behind the sum of solutions, i.e.

for the homogenous: x + y = 0 ; y = -x

  (x_1, -x_1) + (x_2, -x_2)
  = (x_1 + x_2, -x_1 + -x_2)
  = (x_1 + x_2, -(x_1 + x_2))
  = (x, -x)

and for the non-homogenous: x + y = 1 ; y = a - x

  (x_1, a - x_1) + (x_2, a - x_2)
  = (x_1 + x_2, (a - x_1) + (a - x_2))
  = (x_1 + x_2, 2a - x_1 - x_2)
  = (x_1 + x_2, 2a - (x_1 + x_2))
  = (x, 2a - x)
  
but what is the "intuition" behind this?

Is it that all vectors in the solution set of the homogenous equation
lie in the "line" making up the set and thus the sum of two vectors
will also lie in the "line"?

And that adding vectors from the solution set of the non-homogenous
equation will give a vector which is offset from a solutions by the
constant in the equation?

Is this as "intuitive" as this gets or is there some even more
intutive explanation?

> For the second (non-homogeneous) equation, x + y = 1,
> x = 2, y = -1 and x = 3, y = -2 are solutions.
> but x = 2 + 3 = 5, y = -1 + -2 = -3 is not a solution, so that adding
> solutions does not always give a solution.

Hmm, you say "does not always". Does this mean that there are
non-homoegenous equations where two solutions actually add to another
solution?

> Does this simple example help sort things out for you?

Yes, it was a very nice example that got me thinking even harder about
these things!

/Henrik

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