The Basis for a Vector Space
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.