# Thread: Vector space problem (lin. algebra)

1. ## Vector space problem (lin. algebra)

If anyone could explain how the following problem is done, I would really appreciate it!

Here is a vector candidate. The set is R, and we define scalar multiplication by ax = a * x (usual scalar multiplication) and vector addition by x (insert circle symbol with plus sign in it that means vector addition) y = max(x, y).

For each of the following three vector space axioms, either verify the axiom or show that it does not hold.

a) a(x+y) = ax + ay
b) There exists an element 0 such that for any x in the proposed vector space, x + 0 = x.
c) x+y = y+x

2. Originally Posted by buckaroobill
If anyone could explain how the following problem is done, I would really appreciate it!

Here is a vector candidate. The set is R, and we define scalar multiplication by ax = a * x (usual scalar multiplication) and vector addition by x (insert circle symbol with plus sign in it that means vector addition) y = max(x, y).

For each of the following three vector space axioms, either verify the axiom or show that it does not hold.

a) a(x+y) = ax + ay
a(x+y) = a max(x,y)

ax + xy = max(ax,ay)

let a=-1, x=1, y=2, then a(x+y) = -1 max(1,2) = -2, and ax+ay = max(-1,-2) = -1, so
in this case a(x+y) != ax + ay, so this axiom is not satisfied.

b) There exists an element 0 such that for any x in the proposed vector space, x + 0 = x.
If such an element existed then there is a e in R such that for all x in R e<x, but there is
no such element in R (it would have to behave like -infty, but this is not an element of R)

c) x+y = y+x
Trivialy true as max(x,y)=max(y,x)

RonL