Linear Algebra Answers

QUESTION 1 Find the coordinate vector [p(x)]ℬ of p(x) = 5 − x + 3x 2 with respect to the basis ℬ = {u1, u2, u3 } of P2 where u1 = 1 − x + 3x 2 , u2 = 2 and u3 = 3 + x 2

A weight A pound on one side of a beam balances a weight of 40 pounds placed 6 feet from the fulcrum on the other side. If the unknow weight is moved 3 feet nearer the fulcrum, it balances a weight of 20 pounds place 7 1/2 feet from the fulcrum. Find the unknow weight and its distance from the fulcrum in the first instance. (Neglect the weight of the beam.)

A local high school needs to hire several cafeteria workers and bus drivers. Cafeteria workers earn R13 800

per month and bus drivers earn R17 250 per month. The school board gave the school permission to spend

no more than R345 000 on the salaries per month, but must hire more than 15 people.

Let B represent the number of bus drivers hired.

Let C represent the number of cafeteria workers hired.

For every bus driver hired there must be at least two cafeteria workers hired. The number of cafeteria

workers must at most be three more than twice the number of bus drivers.

Write a system of linear inequalities that the school could use to determine the number of cafeteria workers

and bus drivers they can hire

Solve for the determinant in the equation below(10marks)

1.7.1

"\\begin{matrix}\n 4 & -3 & 2 \\\\\n 1 & 2 &-2 \\\\\n 2 & -1 & -4\\\\\n\n\\end{matrix}"

1.7.2.

"\\begin{matrix}\n 2 & -2 & 1 \\\\\n 2 & 2 & 1\\\\\n 4 & 1 & 3 \\\\\n\\end{matrix}"

Solve for the determinant in the equation below. (10)

1.7.1.

4 −3 2

1 2 −2

2 −1 −4

1.7.2

2 −2 1

2 2 1

4 1 3

The vectors V1 - (1, 1, 2, 4), V2 - (2, - I , -5 , 2), V3 = (1, -1 , -4 , 0) and

V4 = (2, 1, 1 ,6 ) are linearly independent. Is it true ? Justify your answer

QUESTION 3 Let A = [ 4 0 1 −2 1 0 −2 0 1 ](its 3by3 matrix).

3.1. Find the eigenvalues of A.

3.2. For each eigenvalue of A find the corresponding eigenvectors.

3.3. Is A diagonalizable? Explain. If it is diagonalizable, then find P such that 𝐷 = 𝑃 −1𝐴𝑃 is a diagonal matrix. Do not compute 𝑃 −1(show all step!) . 

For each of the linear operators below, determine whether it is normal, self adjoint, or neither

(a) T / (R ^ 2) -> R ^ 2 defined by T(x, y) = (2x - 2y, - 2x + 5y)

(b) T / (C ^ 2) -> C ^ 2 defined by T(x,y)=(2x+iu.x+2y)

25. For each of the linear operators below, determine whether it is normal, self adjoint, or neither (a) T : R2 → R2 defined by T(x, y) = (2x − 2y, −2x + 5y). (b) T : C 2 → C 2 defined by T(x, y) = (2x + iy, x + 2y). 

QUESTION 4 Find the quadratic form q(X) that corresponds to the symmetric matrix

𝐴 = [ 1 2 3 2 − 1 2 3 2 − 1] ( is 3by3 matrix)