Dimensions of general vector spaces

The concepts are starting to snowball. I have been studying and doing the homework assigned, hoping that as the concept accumulate it would create a pool of rules that made more sense, but this hasn't worked out. This particular question hit a nerve, because I am unable to make sense of why a linearly independent set can have zero basis and zero dimensions. I kinda understand that there is only the trivial answer to each skalar multiple inorder for the answer to be zero. I do not understand how could a set that is proven to be linear independent, and span R3, have no dimension or basis. Can someone please take a min to explain what is going on?

Find a basis for the solution space of the homogeneous linear system, and find the dimension of that space.

(xn) where n is the subset

2(x1)+(x2)+3(x3)=0

(x1) +5(x3)=0

(x2)+(x3)=0

I augmented the matrix to reduced row echelon form:

1 0 0 0

0 1 0 0

0 0 1 0