Definition · applies to vectors and bases · also called coordinates, components

High school

Consider a set of basis directions, such as “east” and “north.” The coordinate vector of a vector vector $v$ is the list of numbers saying how much of each basis direction is needed to equal $v$.

Undergraduate

Consider a vector space $V$ with a basis $B = (b_1, \dots, b_n)$, and a vector $v$ in $V$, written as $v = c_1 b_1 + \dots + c_n b_n$. The coordinate vector of $v$ is

\[[v]_B = (c_1, c_2, \dots, c_n) \in \reals^n.\]

Properties

  • The coordinate vector depends on the basis; the vector does not.
  • If the columns of $P$ are the new basis vectors written in the old basis, then the new coordinates of $v$ are $P^{-1}$ times its old coordinates.

Distinctions involving this term

  • Vector vs. coordinate vector: A vector is an element of a vector space. Its coordinate vector is a list of numbers (coordinates) that describes it relative to a chosen basis. Changing the basis changes the coordinates, not the vector.

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