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)

1) Find x in the figure below if the perimeter of the parallelogram is 324 feet. Round to the nearest foot.a. 20 ft.b. 10 ft.c. 19 ft.d. 36 ft. 2) The sum of three consecutive even integers is 390. Find the smallest number.a. 128 b. 129c. 130d. 132 3) NVESCO Field at Mile High, home to the Denver Broncos, has 11,675 more seats than Heinz Field, home to the Pittsburgh Steelers. Together, these two stadiums can seat a total of 140,575 NFL fans. How many seats does the Broncos stadium have? (Sources: Denver Broncos, Pittsburgh Steelers)a. 76,125b. 152,250c. 128,900d. 64,450 4) Of the 1376 tornadoes that occurred in the United States during 2003, 39.5% occurred during the Month of May. How many tornadoes occurred during May 2003? Round to the nearest whole.a. 544b. 791c. 543 d. 792 5) One number is five times a first number. A third number is 100 more than the first number. If the sum of the three numbers is 415, find the largest.a. 45b. 145c. 163d. 2256) A flooded piece of land near the Mississippi River in New Orleans is to be surveyed and divided into 4 rectangles of equal dimension as shown below. Express the total amount of fencing needed as an algebraic expression in x.a. x + 3x - 15b. 4x + 5(3x - 15)c. 8x + 5(3x - 15)d. 8x + 3x - 15 7) Which expression could be used to represent the total amount of money (in cents) in x nickels, (x + 3) dimes, and 2x quarters? (Hint: the value of a nickel is 5 cents, the value of a dime is 10 cents, and the value of a quarter is 25 cents).a. x + x + 3 + 2xb. 5x + 10(x + 3) + 25xc. 5x + 10x + 3 + 25(2x)d. 5x + 10(x + 3) + 25(2x) 8) Which of the following is not one of the four steps for problem solving?a. Discoverb. Interpretc. Understandd. Solve 9) Which algebraic expression could be used to find the perimeter of the figure below?a. 3x + x + 3x - 1b. 3x + x + x + 3x - 1c. 3x + 3x + x + xd. 2x + 3x - 1 10) Which of the following algebraic expressions should be used to model the sentence below?The sum of three consecutive odd integers, if x + 1 is the first odd integer.a. x + x + 2 + x + 4b. x + x + 1 + x + 2c. x + 1 + x + 3 + x + 5d. x + 1 + x + 2 + x + 3

n(C)=