In Exercises 4 5 6 and 12 determine whether each set equippe

In Exercises 4, 5, 6, and 12, determine whether each set equipped with the given operation is a vector space. For those that are not vector spaces, identify the vector space axioms that fail

Solution

if u = (x,x,...,x) and v = (y,y,...,y) then:

u+v = (x,x,...,x) + (y,y,...,y) = (x+y,x+y,...,x+y)

which is also of the required form.

and for any scalar c, c(u) = c(x,x,...,x)

= (cx,cx,...,cx), also of the required form.

this shows the set is closed under vector addition and scalar multiplication.

it only remains to show that 0 is also of the required form, which is obvious,

since 0 = (0,0,...,0).