define * on Q by a * b = ab
Transformation (Diagonalizable; Eigenvector; Eigenvalue)
Verify: (a) If A is diagonalizable and B is similar to A then B is also diagonalizable. (b) If {see attachment} and x is an eigenvector of A corresponding to an eigenvalue ... {see attachment for complete question
If there are n vectors v1, v2, v3...vn in E^m, which spans a subspace of dimension k<=n. If k
Transformation - Algebraic Number Theory
Please see the attached problem relating to null space, range, eigenvalues and transformation. Thank you.
Let H = [...] where a,b is an element of C(complex numbers). Is (H, +, .) a field? If not, give reasons. (See attachment for full question)
Let F = [...] where a,b is an element of C(complex numbers), a² + b² ≠ 0. Is (F, +, .) a field? (See attachment for full question)
LET F be a field and set G = {see attachment --> NOTE: a,b is an element of F} Under what conditions on F will G be a field? Can you give an example of such F other than R (real numbers)?
Prove that if G1 and G2 are abelian groups, then the direct product G1 x G2 is abelian.
Let G sub1 and G sub 2 be groups, with subgroups H sub 1 and H sub 2, respectivetly. Show that {(x sub 1, x sub 2) | H sub 1 is an element of H sub 1, x sub 2 is an element of H sub 2} is a subgroup of the direct product G sub 1 x G sub 2.
Find the order if each of these permutations. (1,2)(2,3)(3,4) and (1,2,5)(2,3,4)(5,6)
