Vector vs. coordinate vector

Linear Algebra · Siblings

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.

Definitions

vector

Elementary

A vector is an arrow with a length and a direction.

High school

A vector is a quantity with a magnitude and a direction. Graphically, a vector is drawn as an arrow. In coordinates, it is written as a list of numbers.

Caveat. In general vector spaces, such as spaces of functions or polynomials, there may be no meaningful notion of “magnitude” or “direction” until one is chosen.

Undergraduate

A vector is an element of a vector space: a set with vector addition and scalar multiplication satisfying the vector space axioms. Vectors are defined by what you can do with them, not by what they look like.

Examples

• $(2, 3) \in \reals^2$
  $(2, 3)$ is a vector, because $\reals^2$ has a distinguished origin, $(0, 0)$, so each point can be identified with the vector from the origin.
• $(3, 2)$ is the coordinate vector of $(3, 2) \in \reals^2$, relative to the standard basis
  $(3, 2)$ is a vector in $\reals^2$, because it is an element of the vector space $\reals^2$.
• A velocity, in meters per second
  A velocity is a vector because it is an arrow in space, with units of length per time.
• The polynomial $p(x) = 3 + 2x$
  $p(x) = 3 + 2x$ is a vector in the space of polynomials of degree 1 or less.

Non-examples

• The location of Paris
  The location of Paris is not a vector, because adding the locations of Paris and the location of another city, such as Rome, has no meaning.

coordinate vector

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
  $(3, 2)$ 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 $(1, x)$
  $3 + 2x$ 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 $(1 + x, x)$
  $3 + 2x$ 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

• The polynomial $p(x) = 3 + 2x$
  $3 + 2x$ 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.

Examples

Example	vector	coordinate vector	Why
The polynomial $p(x) = 3 + 2x$	✓	✗
$(3, 2)$ is the coordinate vector of $3 + 2x$, relative to the basis $(1, x)$	–	✓
$(3, -1)$ is the coordinate vector of $3 + 2x$, relative to the basis $(1 + x, x)$	–	✓	The same vector $3 + 2x$ has different coordinates in a different basis.
$(3, 2)$ is the coordinate vector of $(3, 2) \in \reals^2$, relative to the standard basis	✓	✓	In $\reals^n$ with the standard basis, a vector is its own coordinate vector, which is why vectors and coordinate vectors are easy to confuse.

See also

• Vector vs. point
• Vector vs. covector
• Array vs. matrix vs. linear map

All distinctions · All definitions