Linear Algebra Answers

1.   The figure shows two bases, consisting of unit vectors, for ll?²: S —— (i,j) and 0 = (ut,ut).

(a)    Find the transition matrix  s B

(b)    Find the transition matrix PB S

In each part of Exercises 21–22, find the eigenvalues and the

corresponding eigenspaces of the stated matrix operator on R3.

Refer to the tables in Section 4.9 and use geometric reasoning to

find the answers. No computations are needed.

21. (a) Reflection about the xy-plane.

(b) Orthogonal projection onto the xz-plane.

(c) Counterclockwise rotation about the positive x-axis

through an angle of 90◦.

(d) Contraction with factor k (0 ≤ k < 1).

22. (a) Reflection about the xz-plane.

(b) Orthogonal projection onto the yz-plane.

(c) Counterclockwise rotation about the positive y-axis

through an angle of 180◦.

(d) Dilation with factor

Employ the Gauss-Seidel method, solve the system.

10𝑥 + 𝑦 + 𝑧 = 12

2𝑥 + 2𝑦 + 10𝑧 = 14

2𝑥 + 10𝑦 + 𝑧 = 13

Use simplex method to maximize 𝑓 = 3𝑥 + 5𝑦 + 4𝑧 subject to the conditions 2𝑥 + 3𝑦 ≤ 18 2𝑥 + 5𝑦 ≤ 10 3𝑥 + 2𝑦 + 4𝑧 ≤ 15 and 𝑥, 𝑦, 𝑧 ≥ 0.

Use simplex method to maximize 𝑓 = 4𝑥 + 5𝑦 subject to the conditions 𝑥 + 2𝑦 ≤ 5 𝑥 − 2𝑦 ≤ 2 −𝑥 + 𝑦 ≤ 2 2𝑥 + 𝑦 ≤ 6 and 𝑥, 𝑦 ≥ 0.

Show that the set 𝑆 = {(1, 0, 1)(1,1, 0)(−1, 0, −1)} is linearly dependent in 𝑉3(𝑅)

You are given the following matrix

 "A=\\left(\\begin{array}{cccc}a & b & c\\\\ d & e & f \\\\g & h & i\\end{array} \\right)."

Which of the following is the determinant of A?

1. "aei+bfg\u2212cdh\u2212ceg\u2212afh\u2212bdi"
2. "aei+bfg+cdh\u2212ceg\u2212afh+bdi"
3. "aei\u2212bfg+cdh\u2212ceg+afh\u2212bdi"
4. "aei+bfg+cdh\u2212ceg\u2212afh\u2212bdi"
5. "\u2212aei\u2212bfg\u2212cdh+ceg+afh+bdi"

Find all real values of λ such that 

"\\left| \\begin{array}{cccc}1 & \\lambda & -1\\\\ 1 & -1 & -\\lambda \\\\ \\lambda & -1 & 1\\end{array} \\right |=0."

1. λ=0
2. "\\lambda=\\sqrt{3}"
3. "\\lambda=-\\sqrt{3}"
4. There is no such λ
5. λ=1

Consider the linear transformations :

f:R^2→R^2 g:R^2→R^2

f (x,y)→(2x-y,3x+y) g(x,y)→(4x-2y,2x+y)

a) Determine the matrix of linear transformation f relative to the basis

{e1=(1,2),e2=(-1,1)}

b)dertimine (fog)(xy)

c) find g^-1