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:
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)
“Without being forgiven, released from the consequences of what we have done, our capacity to act would ... be confined to one single deed from which we could never recover; we would remain the victims of its consequences forever, not unlike the sorcerer’s apprentice who lacked the magic formula to break the spell.”
—Hannah Arendt (1906–1975)