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
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