Question

The set of all n-tuples of real numbers of the form (x,x,x,......,x)with the standard operations on R^n. is this vector space or not

Answer 1

isnt this the same problem below?

Answer 2

Let V be a set on which addition and scalar multiplication are defined (this means that if u and v are objects in V and c is a scalar then we’ve defined and cu in some way). If the following axioms are true for all objects u, v, and w in V and all scalars c and k then V is called a vector space and the objects in V are called vectors.

(a) is in V This is called closed under addition.
(b) cu is in V This is called closed under scalar multiplication.
(c) u+v=v+u
(d) u+(v+w)=(u+v)+w
(e) There is a special object in V, denoted 0 and called the zero vector, such that for all u in V we have 0*u=0
(f) For every u in V there is another object in V, denoted -u and called the negative of u, such that u+(-u)=0
(g) c(u+v)=cu+cv
(h) (c+k)u=cu+ku
(i) c(ku)=cku
(j) 1u=u
So you have to prove these cold for all n-tuples denoted:
\[X=\left\{ x_1,x_2,...x_{n-1},x_n \right\}\]

Answer 3

(a) u+v is in V**