Chapter 5

Apuntes

Chapter 5 Linear Mappings 5.1 Linear Mappings and Isomorphisms One of the simplest functions of real numbers is a proportional function f (x) = ax (a, x ∈ R). Linear mappings can be considered as a generalization of propor- tional functions to higher-dimensional spaces. Linear mappings Let U and V be vector spaces over K . A mapping T of U into V is called a linear mapping (over K ) when T satisfies the following conditions: (1) T (u + v) = T (u) + T (v) (u, v ∈ U ), (2) T (cu) = cT (u) (u ∈ U, c ∈ K ). When T is a linear mapping of U into V , we denote it by T : U −→ V. Any linear mapping T of U into V maps the zero vector 0U of U to the zero vector 0V of V . In fact, we see that T (0U ) = T (0 · 0U ) = 0 · T (0U ) = 0V . The linear mapping of U into V which maps all vectors in U to the zero vector 0V of V is called the zero mapping. We denote the zero mapping of U into V by OU,V or simply by O. Example 1 Let A be an m × n matrix and TA the mapping of K n into K m defined by TA(x) = Ax (x ∈ K n ). Then TA is a linear mapping of K n into K m . In fact, TA satisfies the two conditions of the definition of linear mapping: © The Author(s), under exclusive license to Springer Nature Si...