1. The number of elements in the basis is equal to the dimension of the space.
2. The set of elements in the basis must be linearly independent.
For example, in
\[\mathbb{R}^3\]
the vectors\[\begin{pmatrix}1\\0\\0\end{pmatrix} , \begin{pmatrix}0\\1\\0\end{pmatrix}, \begin{pmatrix}0\\0\\1\end{pmatrix} \]
are linearly independent, and so are the vectors
\[\begin{pmatrix}1\\1\\0\end{pmatrix} , \begin{pmatrix}0\\1\\-4\end{pmatrix}, \begin{pmatrix}1\\1\\1\end{pmatrix} \]
Both sets have three vectors, equal to the dimension of
\[\mathbb{R}^3\]
and so both these sets form a basis.