First of all, obviously there are inifinite possibilities for bases in every vector space. You can always multiply all of a given basis elements in a scalar and get a new, different, legitimate basis, for example.

However, the dimension of a vector space is defined well. consequently, you and your source cannot both be right, since you found 3 linearly independent vectors which form (in your opinion) a basis, meaning that dim(V) = 3, while according to your source, dim(V) = 2.

Sorry to say that your source is indeed correct. The vector space is spanned by these two vectors: {(1,0,2,0),(0,1,-3,2)}.

That means that each vector, including the others you've written, can be given as a linear combination of these two vectors.

To find that out - write all of the vectors as columns (or rows) in a matrix, and use Gaussian elimination to find the two linear independent vectors I've mentioned.

(Notice that there are other possibilities, but these two are the ones I got as well by elementry actions, and I think so should you.)