/sci/ - Science & Math

A question about vector space, from intro to quantum mechanics Griffiths 2nd Ed.

>If the ordinary vector in 3D is (a_x, a_y, a_z) with complex components:
>a) Does the subset of all vectors with a_z = 0
constitute a vector space? If so what is its dimensions: if not, why not?
My answer: yes, it constitutes a vector space because it can be satisfied through vector addition and scalar multiplication. The dimension of the vector space is 2.

>What about the subset of all vectors whose z component is 1?
My answer: Yes, the subset of vectors whose z component is 1 would be a vector space because they are closed under vector addition and scalar multiplication, work is then shown on paper. The dimension of this vector space would be infinitely many?

>What about the subset of vectors whose components are all equal?
My answer: Yes, if |a> = |b> = ..., then the subset of vectors whose components are all equal constitute a vector space. However, because the vectors are linearly dependent, the dimension of the subset would be 3?

Where am I correct and where did I go wrong, thanks for the help.