In my Algebraic Geometry course, a tangent space at a point $a$ on a variety was defined as the set of lines $L = \{ a + t \alpha | t \in, k\}$ with intersection multiplicity greater than or equal to 2, where $k$ is the field which we are working on. I thought I understood the definition in the class but cannot make sense of it after coming back. For a start how do we define addition on this space so as to make it a vector space? If we think of adding lines as just adding their slopes the problem arises when the same line can be represented by a scalar multiple of the present slope. Can someone help me out here?
PS: Even though I tried to find references for this, most of it uses some definition with sheaves which I am not very familiar with. Is it possible to provide some references for these?( We are presently doing the first chapter in Hartshorne) Thanks in advance.