How many ways are there to define a vectorial subspace?

Question

It is a very simple question, but I've been very confused with teachers defining the vectorial spaces in a lot of ways. Are the following ways to define a vectorial space correct? And are there more ways?

1. With equations $V=\{ \vec{x}\in \mathbb{R}^3 : x_1+x_2-x_3=0; 4x_1+x_2-3x_3=2 \}$
2. With vectors itself $V=\{\vec{x}\in \mathbb{R}^2 : \vec{x}=2\vec{i}-\vec{j}; \vec{i}=(1,0), \vec{j}=(0,1)\}$
3. With spans $V=span \{\vec{u_1},\dots,\vec{u_k}\}$

Note that I know that one can be transformed into the other but that isn't what I'm interested into.