In the definition of a vector space, vector addition x+y =y+x and scalar multipl
ID: 1949197 • Letter: I
Question
In the definition of a vector space, vector addition x+y =y+x and scalar multiplication cx must obey the following 8 rules:(1) x+y=y+x, commutative
(2) x + (y + z) = (x+ y) + z
(3) x + 0 =x for all x
(4) unique -x for each x, s.t. x+(-x) =0
(5) 1 Times x = x
(6) (c1c2)x = c1(c2x)
(7) c(x+y) =cx + cy
(8) (c1 + c2)x = c1x + c2x
Question: If the sum of the "vectors" f(x) and g(x) is defined to be the function f(g(x)), then the "zero vector" is g(x)=x. Keep the usual scalar multiplication cf(x). What are the four rules that are broken?
Explanation / Answer
I assume f and g are defined from R to R 1) is broken because in general fog is different than gof About any choices of f and g will show that 2) this is true because composition is associative f o(g oh)=f o(g oh) 3) true f o Id=f 4) not true because the existence of an inverse would imply the function is bijective But not all functions are bijective 5)true 1*f=f 6) true it is obvious by multiplication with scalars 7) this and 8 are a bit quirky it would mean c*(f og)=(cf)o(cg) it is not true for example f=g=Id, then c*(f og)(x)=cx while (cf)o(cg)(x)=cf(cx)=c^2x 8) it would mean (c1+c2)f=(c1f)o(c2f) not true for example f=Id (c1+c2)f(x)=(c1+c2)x (c1f)o(c2f)(x)=(c1 f)(c2 x)=c1 c2 x