Linear map ↩

Metadata 
shorthands: {}
aliases: [Vector space homomorphism, Homomorphism of vector spaces, Linear transformation]
created: 2021-11-06 12:22:27
modified: 2022-01-10 04:13:04

are vector spaces over the same field .
is a linear map if it respects linear combinations, i.e.:

Identity map

The identity map is for which for every .

Examples

  1. Suppose is linear. Then is completely determined by knowing .
    (Proof: If , then )
  2. The previous is also true for any linear map
  3. The functions of differentiability class defined on form a vector space, let's call it . Then we have a map

    defined by letting

    (maps to the derivative of the input function) Then is a linear map.
  4. Have the same as in the previous, we define

    by . Then is linear.