Proof : Diagonalization of Matrices

Please see the attached file for full problem description. Write a proof for the following statement: If A is an n x n upper triangular matrix with no two diagonal elements the same, then A is similar to a diagonal matrix. Show work.

Findin the Equation of a Reflecting Line

Determine if the following orthogonal matrix represents a rotation or a reflection of the plane with respect to the standard basis. Find the equation of the reflecting line. - - |3/5 4/5 | |4/5 -3/5 | - -

Idempotent linear transformation

A linear transformation L:V->V is said to be idempotent if L dot L = L. If L is idempotent, show that there exists a basis S={a1,a2,...,an} for V such that L(ai)=ai for i= 1,2,...,r and L(aj) = 0v for j= r+1,...,n, where r= p(L). Describe the matrix representing L with respect to the basis S.

Nilpotent transformation

Consider the transformation N: V->V. Let g be a vector such that N^k-1 does not equal 0, but N^k = 0. First show that the vectors g,N(g),N^2(g),..,N^k-1(g) are linearly independent, and then (assuming V has dimension n) If N is nilpotent of index n, show that the set S= {g, N(g), N^2(g),...,N^n-1(g)}is a basis for V. Describe the matrix which represents N with respect to the basis S.

Linear algebra; linear dependence

Please see the attached file for full problem description.