[Nomanirvana]

Hi -- I'm studying introductory linear algebra and doing some C# programming along with it to solidify it in my mind.

I'm wondering if its possible to create an algorithm that would determine if some set is a subspace of R^n (ie a Euclidian vector space).

For example, consider the set

A = { (x, 3x + 1) : x e R }

To determine if its euclidian it has to be both closed under addition and scalar multiplication.

So, in order for a vector v = (v1, v2) to be in A, the second component v2 must be 1 greater than three times the first component v1.

Two points in this system are (1,4) and (2,7) -- the sum of which is (3, 11). However, (3, 11) does not satisfy the set, so this is not closed under addition and is not a subspace.

An algorithm to determine this could simply check and see if all possible sums satisfy the set, but that would be cumbersome. I'm looking for an alternative. It seems to me that any time constants are involved in the elements of the set that would likely throw off the whole closed under addition thing, but I'm not sure.

This post has been edited by Nomanirvana: 22 November 2010 - 12:25 AM