A [[basis for vector space|basis]] $\set{v_1\ldots v_n}$ generates a [[vector space]] $V$, over [[field]] $F$ which also contains a [[vector]] $w$ that can be represented using $x_1 \ldots x_n$ $ w=x_{1} v_{1}+\cdots+x_{n} v_{n} $ Then $X=(x_1, \ldots, x_n)$ will be called the **coordinate**, or *coordinate vector*, with respect to $V$,