the "naive" way is to ask yourself: how many complex numbers do i need to specify to identify my "vector"? for 3x2 matrices, that number is 6 (one for each entry).

however, mathematics professors being what they are, will usually insist you display a basis (a linearly independent spanning set). can you think of a set of six matrices,

each of which captures the idea of "a single matrix coordinate"? once you have done this, show linear independence and spanning.

again, with any vector space V, there are 3 things you need to show for any subset W:

1) W is non-empty (preferrably by showing the 0-element is a member. if the 0-element (0-vector) is not in W, you will not obtain a vector space).

2) if u,v are in W then their vector sum u+v must also be.

3) if c is any scalar in your underlying field, and u is in W, then cu must also be in W. be careful with this one. if you are working with a complex vector space, it is not sufficient to check this property for real scalars only.

attempt to show these properties (or give a counter-example) for each of the sets in your post. that's how it's done.