Linear mapping

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In mathematics, a linear mapping (or linear transformation) is a mapping f between vector spaces that preserves addition and scalar multiplication.[1][2][3]

Definition[change | change source]

Let V and W be vector spaces over the same field K. A function f: VW is said to be a linear mapping if for any two vectors x and y in V and any scalar α in K, the following two conditions are satisfied:

f(\mathbf{x}+\mathbf{y}) = f(\mathbf{x})+f(\mathbf{y}) \!
f(\alpha \mathbf{x}) = \alpha f(\mathbf{x}) \!


Sometimes a linear mapping is called a linear function.[4] However in basic mathematics, a linear function means a function whose graph is a line.

See also[change | change source]

References[change | change source]

  1. Lang, Serge (1987). Linear algebra. New York: Springer-Verlag. p. 51. ISBN 9780387964126 .
  2. Lax, Peter (2007). Linear Algebra and Its Applications, 2nd ed.. Wiley. p. 19. ISBN 978-0471-7516=56-4 . (English)
  3. Tanton, James (2005). Encyclopedia of Mathematics, Linear Transformation. Facts on File, New York. p. 316. ISBN 0-8160-5124-0 . (English)
  4. Sloughter, Dan (2001). "The Calculus of Functions of Several Variables, Linear and Affine Functions" (in English). http://cfsv.synechism.org/c1/sec15.pdf. Retrieved February 2014.