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

Vector Space : Linear Dependence and Null Space

If there are n vectors v1, v2, v3...vn in E^m, which spans a subspace of dimension k<=n. If k

Transformation : Null Space, Range and Eigenvalues

Please see the attached problem relating to null space, range, eigenvalues and transformation.

Fields : Complex Numbers

Please see the attached file for the fully formatted problems. LET H = a b -b a : a,b is an element of C (complex numbers) Is (H, +, .) a field? If not, give reasons.

Fields : Complex Numbers

LET F = a b -b a : a,b is an element of C (complex numbers), a² + b² ≠ 0 Is (F, +, .) a field?

Fields

LET F be a field and set G = a b -b a : 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)?

Abelian Groups

Prove that if G1 and G2 are abelian groups, then the direct product G1 x G2 is abelian.

Groups and Subgroups

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.

Permutation : Order

Find the order if each of these permutations. (1,2)(2,3)(3,4) and (1,2,5)(2,3,4)(5,6)

Pemutation Groups : Order of Rigid Motions of a Cube

Find the order of the group of rigid motions of a cube.