Mathematics Homework Help

this question deals with the “Laguerre Polynomials,” which are orthogonal
with respect to the inner product hf, gi =
Z ∞
0
f(x)g(x) e
−x
dx. Use the fact that
∀ integers p, q ≥ 0 hx
p
, xq
i = (p + q)! (“p plus q factorial”), to derive the first 4 (order
0, 1, 2, 3) orthonormal Laguerre Polynomials starting from elements of the standard
polynomial basis {1, x, x2
, x3}.

Let T ∈ L(C
7
) be defined by
T(z1, z2, z3, z4, z5, z6, z7) = (πz1+z2+z3+z4, πz2+z3+z4, πz3+z4, πz4,
√
7z5+z6+z7,
√
7z6+z7,
√
7z7)
Let Bs(C
7
) = {e1, e2, e3, e4, e5, e6, e7} be the standard basis of C
7
(a) (25 pts.) Find M(T, Bs(C
7

(b) (25 pts.) Find the eigenvalues {λk}k=1,…,?
(c) For each eigenvalue, λk:
i. (30 pts.) Find the eigenspace E(λk, T)
ii. (30 pts.) Find the generalized eigenspace G(λk, T)