## The Basis for a Vector Space

A basis for a vector space is the smallest set of vectors that spans the space, so that any vector in the space can be written as a linear combination of elements of the basis. A basis for a vector space is not unique, but any basis must satisfy the following requirements1. 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.