Linear mapping

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

Let V and W be vector spaces over the same field K. A function f: V → W 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.

References

↑ Lang, Serge (1987). Linear algebra. New York: Springer-Verlag. p. 51. ISBN 9780387964126.
↑ Lax, Peter (2007). Linear Algebra and Its Applications, 2nd ed. Wiley. p. 19. ISBN 978-0471-7516=56-4. {{cite book}}: Check |isbn= value: invalid character (help) Template:En
↑ Tanton, James (2005). Encyclopedia of Mathematics, Linear Transformation. Facts on File, New York. p. 316. ISBN 0-8160-5124-0. Template:En
↑ Sloughter, Dan (2001). "The Calculus of Functions of Several Variables, Linear and Affine Functions" (PDF). Retrieved February 2014. {{cite web}}: Check date values in: |accessdate= (help)

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[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. {{cite book}}: Check |isbn= value: invalid character (help) Template:En

[Tanton-3] Tanton, James (2005). Encyclopedia of Mathematics, Linear Transformation. Facts on File, New York. p. 316. ISBN 0-8160-5124-0. Template:En

[4] Sloughter, Dan (2001). "The Calculus of Functions of Several Variables, Linear and Affine Functions" (PDF). Retrieved February 2014. {{cite web}}: Check date values in: |accessdate= (help)

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Definition

See also

References