Linear Algebra and Its Applications 5th. Edition by David C. Lay, Steven R. Lay, Judi J. McDonald | 9780321982384 | 032198238X

Linear Algebra and Its Applications [EXP-40583]

348 SOLVED PROBLEMS

Question: 1.3.7

A company manufactures two products. For $1.00 worth of product B, the company spends $.45 on materials, $.25 on labor, and $.15 on overhead. For $1.00 worth of product C, the company spends $.40 on materials, $.30 on labor, and $.15 on overhead. Let b = [.45 .25 .15] and c = [.40 .30 .15] Then b

Verified Answer:

a. Compute 100b = 100

\left [ \begin{matr...

Question: 2.2.3

A horizontal elastic beam is supported at each end and is subjected to forces at points 1, 2, and 3, as shown in Figure 1. Let f in R^3 list the forces at these points, and let y in R^3 list the amounts of deflection (that is, movement) of the beam at the three points. Using Hooke’s law from

Verified Answer:

Write

I_{3}

= [

e_{1} ...

Question: 2.4.5

A matrix of the form A = [A11 A12 0 A22] is said to be block upper triangular. Assume that A11 is p × p, A22 is q × q, and A is invertible. Find a formula for A^-1

Verified Answer:

Denote

A^{-1}

by B and partition...

Question: 5.7.PP.1

A real 3 × 3 matrix A has eigenvalues -.5, .2 + .3i, and .2 – .3i, with corresponding eigenvectors v1 = [1 -2 1], v2 = [1+2i 4i 2], and v3 = [1-2i -4i 2] 1. Is A diagonalizable as A = PDP^-1, using complex matrices? 2. Write the general solution of x’ = Ax using complex eigenfunctions, and then

Verified Answer:

1. Yes, the 3 × 3 matrix is diagonalizable becaus...

Question: 4.6.5

A scientist has found two solutions to a homogeneous system of 40 equations in 42 variables. The two solutions are not multiples, and all other solutions can be constructed by adding together appropriate multiples of these two solutions. Can the scientist be certain that an associated

Verified Answer:

Yes. Let A be the 40 × 42 coefficient matrix of th...

Question: 1.5.2

A single linear equation can be treated as a very simple system of equations. Describe all solutions of the homogeneous “system” 10×1 – 3×2 – 2×3 = 0 (1)

Verified Answer:

There is no need for matrix notation. Solve for th...

Question: 8.6.PP.1

A spline usually refers to a curve that passes through specified points. A B-spline, however, usually does not pass through its control points. A single segment has the parametric form x(t) = 1/6[(1 – t)^3p0 + (3t^3 – 6t^2 + 4)p1 + (-3t^3 + 3t^2 + 3t + 1)p2 + t^3p3] (14) for 0 ≤ t ≤ 1, where p0, p1

Verified Answer:

1. From equation (14) with

t = 0 , x(...

Question: 2.5.3

a. Compute the transfer matrix of the ladder network in Figure 4. b. Design a ladder network whose transfer matrix is [ 1 -8 -.5 5] .

Verified Answer:

a. Let

A_{1}

and

A_{2} ...

Question: 4.6.3

a. If A is a 7 × 9 matrix with a two-dimensional null space, what is the rank of A? b. Could a 6 × 9 matrix have a two-dimensional null space?

Verified Answer:

a. Since A has 9 columns, (rank A) + 2 = 9, and h...

Question: 4.1.PP.3

An n × n matrix A is said to be symmetric if A^T = A. Let S be the set of all 3 × 3 symmetric matrices. Show that S is a subspace of M3×3, the vector space of 3 × 3 matrices.

Verified Answer:

The subset S is a subspace of

M_{3 \times ...