Prove or disprove that the following is a vector space or not:

A.

the set {(x. y): x >= 0, y is a real number} with standard operations in R2.

let r = -1 and (x,y) = (1, 2)

then r(x, y) = (-1, -2), but x must be >= 0 (not sure about this part).

thus this is not a vector space.

B.

the set {(x,x): x is a real number} with standard operations in R2.

let (x1, x1) and (x2, x2) be an element of R2

then (x1, x1)+ x2, x2) = (x1 + x2, x1 + x2), which is an element of R2

thus closed under addition

let r be any real number and (x1, x1) be an element of R2

the, r(x1, x1) = (rx1, rx1) which is an element of R2

thus closed under scaler multiplication

thus this is a vector space

C. the set of all continuous functions defined on the interval [0, 1] with standard operations.

I thought this was false, but the book says its true.

My reasoning for this is because it is possible that if two functions are both on the interval [0, 1] and you add them up, then it is also possible that it may exceed the interval of [0, 1], ie [0,1] + [0,1] = [0,2]. Or am I thinking about this totally wrong?

This post has been edited by **streek405**: 28 March 2015 - 01:10 PM