subspace vs convex cone

[AE lover]

Hi all,

I am confused. What is the difference between subspace and convex
cone. From their definitions, I don't see any difference.

subspace: a set of vector which is closed with addition and scalar
multiplication. That means if S is a subspace, x, and y belongs to s,
we have x*a+y*b belongs to S with all scalars a and b.

convex cone: I see the similarity.

Please tell me the differences.

Thanks

[Paul Sperry]

In article
<dde9ee94-4554-458e...@y38g2000hsy.googlegroups.com>, AE
lover <aelo...@gmail.com> wrote:

The difference is that for the convex cone you only need the much
weaker property that if x an y are in the cone and a and b are
_positive_ (or non-negative depending on who is doing the defining)
then a*x + b*y is in the cone. For example, -x need not be in the cone.

In R^2, think of two half-lines starting at the origin. The pie shaped
area consisting of the lines and the area between them is a classic two
dimension convex cone. As you know, I hope, the smallest _subspace_
containing those two lines is all of R^2.

--
Paul Sperry
Columbia, SC (USA)

[AE lover]

Oh, yeah, I forgot that in the definition of a convex cone, a and b
should be positive. Thanks.

. But I am still confused
On Aug 5, 11:37 pm, Paul Sperry <plspe...@sc.rr.com> wrote:
> In article
> <dde9ee94-4554-458e-8756-27fabd6ad...@y38g2000hsy.googlegroups.com>, AE

[Paul Sperry]

In article
<61ecd014-b305-4eaa...@l64g2000hse.googlegroups.com>, AE
lover <aelo...@gmail.com> wrote:

[Top posting fixed.]

What's the problem?

Here's another example:

Look at the first quadrant in R^2: { (x, y) : x > 0, y > 0}. Clearly,
this is a convex cone since if (u, v) and (r, s) are in the first
quadrant and a and b are positive then a(u, v) + b(r, s) will have
positive coordinates and thus will be in the first quadrant.

On the other hand, if the first quadrant was a _subspace_ then, since
(1, 1) is in the first quadrant then -2(1, 1) would be too. But it's
not.