Linear Map - Definition and First Consequences

Definition and First Consequences

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

additivity
homogeneity of degree 1

This is equivalent to requiring the same for any linear combination of vectors, i.e. that for any vectors x1, ..., xm ∈ V and scalars a1, ..., am ∈ K, the following equality holds:

Denoting the zeros of the vector spaces by 0, it follows that f(0) = 0 because letting α = 0 in the equation for homogeneity of degree 1,

f(0) = f(0 ⋅ 0) = 0 f(0) = 0.

Occasionally, V and W can be considered to be vector spaces over different fields. It is then necessary to specify which of these ground fields is being used in the definition of "linear". If V and W are considered as spaces over the field K as above, we talk about K-linear maps. For example, the conjugation of complex numbers is an R-linear map C → C, but it is not C-linear.

A linear map from V to K (with K viewed as a vector space over itself) is called a linear functional.

These statements generalize to any left-module RM over a ring R without modification.

Read more about this topic:  Linear Map

Famous quotes containing the words definition and/or consequences:

    “Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
    —The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)

    “There is a delicate balance of putting yourself last and not being a doormat and thinking of yourself first and not coming off as selfish, arrogant, or bossy. We spend the majority of our lives attempting to perfect this balance. When we are successful, we have many close, healthy relationships. When we are unsuccessful, we suffer the natural consequences of damaged and sometimes broken relationships. Children are just beginning their journey on this important life lesson.”
    —Cindy L. Teachey. “Building Lifelong Relationships—School Age Programs at Work,” Child Care Exchange (January 1994)