Math for Non-Geeks/Exercises Linear Maps

oben We have compiled some tasks on linear maps here. The proof structures can help you to solve other similar tasks. As a reminder, here is the definition of a linear map: Linear map Let be a mapping between the two vector spaces and . We call a linear map from to if the following two properties are satisfied: ordered Additivity: For all we have Homogeneity: For all and we have Proving linearity of a mapping Linear maps from to Linear map into a field Let be defined by . Show that the map is linear. First you have to show the additivity and then homogeneity of the map. Additivity For this step, let and . Thus is additive. Homogeneity Let and . Thus is homogeneous and is linear. Linear map from to Lineare_Abbildung_von_3D_nach_2D Show that the map with is linear. You have to show that for and it holds true that And you have to show that for , it holds true that Aktuelles Ziel Additivity Aktuelles Ziel Scaling Linearity of the embedding Show that for , the map is linear. Let and , as well as . By definition of the map , we have that So is linear. aufgabe_linearität_R^2 Important special cases aufgabe_identität_linear aufgabe_nullabbildung_linear Linear maps between function spaces Mapping on a function space Consider the function space of all functions from to , as well as the map Show that is linear. The operations on the function space are defined element-wise in each case. That means: for , and we have that and . In particular, this is true for , which implies and Thus, we have established linearity. präkomposition_linear folgenraum_abbildung_linear Construction of a linear map from given values Construction of a linear map Let . Further, consider . Find a linear map with for all . Hint: Use the principle of linear continuation. We see that is a basis of , namely the standard basis. According to the theorem of linear continuation, we can construct a linear map defined by Now we only have to check if is satisfied. It is true that , so Thus, the condition is satisfied for each . The mapping is linear by definition, so we are done. konstruktion_lin_Abb Linear independence of two preimages Let be a linear map and let and be two distinct vectors from , both mapped to a vector with . Prove that and are linearly independent. We show that the two vectors cannot be linearly dependent. So assume that were linearly dependent. Then there would be a such that . We now map these two dependent vectors into the vector space using the linear map . This yields Since by premise, , this is a contradiction and our assumption of linear dependence must be false. Assume that and were linearly dependent. Then there would be a with and . Since the map is linear, it follows that Thus Since by assumption , we must have . But this contradicts our assumption . Thus, we get a contradiction to our assumption of linear dependence. So the vectors are linearly independent. Exercises: Isomorphisms aufgaben_isomorphismus Exercises: Images zuordnung_abbildung_bild bild_einer_matrix surjektivität_dimension Exercises: Kernel aufgabe_kern_bestimmen kern_nilpotenter_endo injektivität_und_dimension unten