Coordinate vector

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

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

Examples

$(3, 2)$ is the coordinate vector of $(3, 2) \in \reals^2$, relative to the standard basis High school
$(3, 2)$ is its own coordinate vector relative to the standard basis, because $(3, 2) = 3 (1, 0) + 2 (0, 1)$.
$(3, 2)$ is the coordinate vector of $3 + 2x$, relative to the basis $(1, x)$ Undergraduate
$3 + 2x$ is the linear combination of the basis vectors using the coefficients $3$ and $2$: $3 + 2x = 3(1) + 2(x)$
$(3, -1)$ is the coordinate vector of $3 + 2x$, relative to the basis $(1 + x, x)$ Undergraduate
$3 + 2x$ is the linear combination of the basis vectors $1 + x$ and $x$ using the coefficients $3$ and $-1$: $3 + 2x = 3(1 + x) + (-1)(x)$

Non-examples

Things that may be mistaken for examples.

The polynomial $p(x) = 3 + 2x$ High school
$3 + 2x$ is not a coordinate vector, because it is not a list of numbers.

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: Vectors in a vector space are often described by lists of numbers, which depend on a choice of basis. 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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