In mathematics, many terms are similar but not interchangeable. This dictionary groups easily confused terms, defines each one, and explains the distinction with examples to illustrate the difference. This work was inspired by the Dictionary of Fine Distinctions: Nuances, Niceties, and Subtle Shades of Meaning by Eli Burnstein.

See also the lists of all definitions and all examples.

Distinctions by subject

Sets and Logic

  • Disjoint union: union of disjoint sets vs. tagged union Conflicting definitions
    “Disjoint union” can mean an ordinary union of sets that happen to be disjoint, or a construction that tags each element with the set it came from, so that copies of overlapping sets are kept apart.

Analysis

  • Types of multivariable derivatives Siblings
    All of these package the partial derivatives of a function differently: as a vector (gradient), a matrix (Jacobian), a linear map (differential), or a single number in a chosen direction (directional derivative) or along a vector field (Lie derivative).

Linear Algebra

  • Array vs. matrix vs. linear map Siblings
    An array is a grid of data. A matrix is a grid of numbers with matrix arithmetic. A linear map is a function between vector spaces. A matrix represents a linear map as an array of scalars with respect to a given vector basis.
  • Vector vs. coordinate vector 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.
  • Vector vs. covector Siblings
    A covector is a linear function that takes a vector and returns a number. In coordinates, both look like lists of numbers, but when the basis changes, their coordinates change in opposite ways.
  • Vector vs. point Siblings
    A point is a location. A vector is a displacement, defined by the operations of addition and scaling. The difference of two points is a vector. Two vectors can be added, and a vector can be added to a point, but the sum of two points is not meaningful.

Geometry

  • Vector vs. point Siblings
    A point is a location. A vector is a displacement, defined by the operations of addition and scaling. The difference of two points is a vector. Two vectors can be added, and a vector can be added to a point, but the sum of two points is not meaningful.

Computing

  • Array vs. matrix vs. linear map Siblings
    An array is a grid of data. A matrix is a grid of numbers with matrix arithmetic. A linear map is a function between vector spaces. A matrix represents a linear map as an array of scalars with respect to a given vector basis.