Question

Linear Algebra

Check whether the following maps are linear. Determine, in the cases that the map is linear, the null space and the range and verify the dimension theorem 1 a. A: R2R2 defined by A(r1, r2r2, xi), b. A: R2R defined by A(z,2)2 c. A: Сз-+ C2 defined by A(21,T2, x3)-(a + iT2,0), d. A: R3-R2 defined by A(r, r2, r3) (r3l,0), C 1

Answer 1

Let Alp?→ R 2 defined by ACX, 2. Null space-t A- ?(x,y) c = 900 10) Nalls pace Ay dimensio) theoreb CNull space ) t to Dimensloo at eRange A) dim Range

t AC1,1): 1-1 스 c. A, C,e? defined by 3 кес. .. At's lined γ . Thcus din Alul)spe 2

By dimension theo rem dim (03)-dim aINull,space, At dimCa angeわ A s not linea -'. A i's not linear.