Which of the following spaces are vector spaces Explain why

Which of the following spaces are vector spaces? Explain why or why not.

The set V = C (the set of complex numbers), and as set of scalars we take again the set of complex numbers C. This time, as + we consider the usual addition of complex numbers, and we define the scalar multiplication · as

· z = ^2 z

for any and z in C.

Solution

given and z in C.

Let\'s take =-1 and z=3 which are clearly in C.

Now plug it into given equation · z = ^2 z

(-1) * 3=(-1)^2*3
-3=3

which is false.

That means given set V doesn\'t satisfy closure property of Vector space which says \"If v in any vector in V, and c is any real number, then the product c · v belongs to V. \"

Hence V is not Vector space.