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:
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)
“[As teenager], the trauma of near-misses and almost- consequences usually brings us to our senses. We finally come down someplace between our parents’ safety advice, which underestimates our ability, and our own unreasonable disregard for safety, which is our childlike wish for invulnerability. Our definition of acceptable risk becomes a product of our own experience.”
—Roger Gould (20th century)