Math for Non-Geeks/Proofs for linear maps

oben We will give here a proof structure that shows how to prove linearity of a map. General procedure Recap: Definition of a linear map We recall that a linear map (or homeomorphism) is a structure-preserving map of a -vector space into a -vector space . That is, for the map , the following two conditions must hold: must be additive, i.e., for we have that: must be homogeneous, i.e., for we have that: . So for a linear map it doesn't matter if we first do the addition or scalar multiplication in the vector space and then map the sum into the vector space , or first map the vectors into the vector space and perform the addition or scalar multiplication there, using the images of the map. Proving that a map is linear The proof that a map is linear can be done according to the following structure. First, we assume that a map is given between vector spaces. That is, and are -vector spaces and is well-defined. Then for the linearity of we have to show: additivity: homogeneity: Introductory example We consider the following map and show that it is linear. First, and are vector spaces over the field . Moreover, the map is well-defined. proving additivity Let . Thus, we have proved the additivity of . proving homogeneity Let and . Then, we have Thus, we have proved the homogeneity of . The map to zero The map to zero is the map which sends every vector to zero. For instance, the map to zero of to looks as follows: The map to zero linear Show that the map is linear. We already know that and are both -vector spaces, and that the map to zero is well-defined. additivity For all we have that homogeneity For all we have that Thus, the map to zero is linear. An example in We consider an example for a linear map of to : with Linearity of Show that the map is linear. is an -vector space. In addition, the map is well-defined. additivity Let and be any vectors from the plane . Then, we have: homogeneity Let and . Then: Thus, the map is linear. A linear map in the vector space of sequences Next, we consider the space of all sequences of real numbers. This space is infinite-dimensional, because there are not finitely many sequences generating this sequence space. But it is a vector space, as we have shown in the chapter about sequence spaces. Sequence space Let be the -vector space of all real-valued sequences. Show that the map is linear. To show linearity, two properties need to be checked: is additive: for all is homogeneous: for all and The vectors and are sequences of real numbers, i.e. they are of the form and with for all . additivity Let and . Then, we have It follows that is additive. homogeneity Let and . Then, we have So is homogeneous. Thus, it was proved that is a -linear map. Abstract example In this chapter, we deal with somewhat more abstract vectors. Let be arbitrary sets; a field and a -vector space. We now consider the set of all maps/ functions of the set into the vector space and denote this set with . Furthermore, we also consider the set of all maps of the set into the vector space and denote this set with . The addition of two maps is defined for by Die scalar multiplication is defined for via Analogously, we define addition scalar multiplication for . The set is a -vector space Show that is a -vector space. Simply check the vector space axioms. We now show that the composition with a mapping is a linear map from to . The composition with a map is linear. Let be a vector space, let be sets, and let or be the vector space of functions from or to . Let be arbitrary but fixed. We consider the mapping Show that is linear. It is important that you exactly follow the definitions. Note that is a map that assigns to every map of to a map of to . These maps, which are elements of and respectively, need not themselves be linear, since there is no vector space structure on the sets and . In order to prove the linearity of , we need to check the two properties again: is additive: for all is homogeneous: for all and So at both points an equivalence of maps is to be shown. For this, we evaluate the maps at every m element . Let . additivity For all we have that Thus, we have shown , i.e., is additive. Let and . homogeneity For all we have that Thus, we have shown , i.e., is homogeneous. Now, additivity and homogeneity of implies that is a linear map. unten