Linear Algebra · Siblings

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 point of $\reals^2$, and it is a vector, because $\reals^2$ has a distinguished origin, $(0, 0)$, so each point can be identified with the vector from the origin.
  • A velocity, in meters per second
    A velocity is a vector because it is an arrow in space, with units of length per time. It is not a covector, because it does not take a vector and return a number.
Non-examples
  • The location of Paris
    The location of Paris is a point. It 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

\[[v]_B = (c_1, c_2, \dots, c_n) \in \reals^n.\]
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
Missing example: the_polynomial_p_x_eq_3_plus_2x
Missing example: 3_2_relative_to_the_basis_1_x
Missing example: 3_minus_1_relative_to_the_basis_1_plus_x_x
Missing example: 3_2_in_reals_squared_relative_to_the_standard

See also

All distinctions · All definitions