New Semester Started Get 50% OFF Study Help! --h --m --s Claim Now

Question Answers

Textbooks

Find textbooks, questions and answers

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Study Help
Tutors
AI Study Planner NEW
Sell Books

Search
Sign In
Register

study help
mathematics
linear algebra

Linear Algebra and Its Applications 4th edition David C. Lay - Solutions

Denote the columns of A by a1, a2, a3, and let W = Span {a1, a2, a3}.LetAnd a. Is b in {a1, a2, a3}? How many vectors are in {a1, a2, a3}? b. Is b in W? How many vectors are in W? c. Show that a1 is in W. [Hint: Row operations are unnecessary.]
let W be the set of all linear combinations of the columns of ALetLet . a. Is b in W? b. Show that the second column of A is in W.
A mining company has two mines. One day's operation at mine #1 produces ore that contains 30 metric tons of copper and 600 kilograms of silver, while one day's operation at mine #2 produces ore that contains 40 metric tons of copper and 380 kilograms of silver. LetAnd Then v1 and v2 represent the
A steam plant burns two types of coal: anthracite (A) and bituminous (B). For each ton of A burned, the plant produces 27.6 million Btu of heat, 3100 grams (g) of sulfur dioxide, and 250 g of particulate matter (solid-particle pollutants). For each ton of B burned, the plant produces 30.2 million
m = m1 + ( ( ( + mkThe center of gravity (or center of mass) of the system is
In Exercises 1 and 2, display the following vectors using arrows on an xy-graph: u, v, -v, -2v, u + v, u - v, and u - 2v. Notice that u - v is the vertex of a parallelogram whose other vertices are u, 0, and -v.1. u and v as in Exercise 12. u and v as in Exercise 2
Let v be the center of mass of a system of point masses located at v1( ( ( ( ( vk as in Exercise 29. Is v in Span {v1( ( ( ( ( vk}? Explain.
A thin triangular plate of uniform density and thickness has vertices at V1 = (0, 1), v2 = (8, 1), and v3 = (2, 4), as in the figure below, and the mass of the plate is 3 g.a. Find the (x, y)-coordinates of the center of mass of the plate. This "balance point" of the plate coincides with the center
Consider the vectors v1, v2, v3, and b in, shown in the figure. Does the equation x1v1 + x2v2 + x3v3 = b have a solution? Is the solution unique? Use the figure to explain your answers.
a. (u + v) + w = u + (v + w)b. c(u + v) = cu + cv for each scalar c
a. u + (-u) = (-u) + u = 0b. c(du) = (cd)u for all scalars c and d
In Exercises 1 and 2, write a system of equations that is equivalent to the given vector equation.1.2.
1. Vectors a, b, c, and d2. Vectors w, x, y, and z
In Exercises 1 and 2, write a vector equation that is equivalent to the given system of equations.1.2.
Compute the products in Exercises 1-2 using (a) the definition, as in Example 1, and (b) the row-vector rule for computing Ax. If a product is undefined, explain why.1.2.
Given A and b in Exercises 1 and 2, write the augmented matrix for the linear system that corresponds to the matrix equation Ax = b. Then solve the system and write the solution as a vector. 1. 2.
Repeat the requests from Exercise 15 with
Exercises 1-2 refer to the matrices A and B below. Make appropriate calculations that justify your answers and mention an appropriate theorem.
In Exercises 1 and 2, mark each statement True or False. Justify each answer. 1. a. The equation Ax = b is referred to as a vector equation.
Use this fact (and no row operations) to find scalars c1, c2, c3, such that
Rewrite the (numerical) matrix equation below in symbolic form as a vector equation, using symbols v1, v2, ( ( ( for the vectors and c1, c2, ( ( ( for scalars. Define what each symbol represents, using the data given in the matrix equation.
Let q1, q2, q3, and v represent vectors in, and let x1, x2, and x3 denote scalars. Write the following vector equation as a matrix equation. Identify any symbols you choose to use. x1q1 + x2q2 + x3q3 = v
Construct a 3 ( 3 matrix, not in echelon form, whose columns span. Show that the matrix you construct has the desired property.
Construct a 3 ( 3 matrix, not in echelon form, whose columns do not span. Show that the matrix you construct has the desired property.
Let A be a 3 ( 2 matrix. Explain why the equation Ax = b cannot be consistent for all b in . Generalize your argument to the case of an arbitrary A with more rows than columns.
Could a set of three vectors in span all of? Explain. What about n vectors in when n is less than m?
Suppose A is a 4 ( 3 matrix and b is a vector in with the property that Ax = b has a unique solution. What can you say about the reduced echelon form of A? Justify your answer.
Let A be a 3 ( 4 matrix, let v1 and v2 be vectors in, and let w = v1 + v2. Suppose v1 = Au1 and v2 = Au2 for some vectors u1 and u2 in. What fact allows you to conclude that the system Ax = w is consistent? {u1 and u2 denote vectors, not scalar entries in vectors.)
Let A be a 5 ( 3 matrix, let y be a vector in, and let z be a vector in. Suppose Ay = z. What fact allows you to conclude that the system Ax = 5z is consistent?
Suppose A is a 4 ( 4 matrix and b is a vector in with the property that Ax = b has a unique solution. Explain why the columns of A must span.
1.
Compute the determinants in Exercises 1-2 using a cofactor expansion across the first row. In Exercises 1-2, also compute the 3. determinant by a cofactor expansion down the second column.1.2.
The expansion of a 3 ( 3 determinant can be remembered by the following device. Write a second copy of the first two columns to the right of the matrix, and compute the determinant by multiplying entries on six diagonals:Add the downward diagonal products and subtract the upward products. Use this
In Exercises 1-2, explore the effect of an elementary row operation on the determinant of a matrix. In each case, state the row operation and describe how it affects the determinant.1.2.
Compute the determinants of the elementary matrices given in Exercises 1-2. (See Section 2.2.)1.2.
1. What is the determinant of an elementary row replacement matrix? 2. What is the determinant of an elementary scaling matrix with k on the diagonal?
a. An n ( n determinant is defined by determinists of (n - 1) ( (n - 1) submatrices? b. The (i, j)-cofactor of matrix A is the matrix Aij obtained by deleting from A its I th row and j th column?
The area of the parallelogram determined byAnd 0 is 6, since the base of the parallelogram has length 3 and the height of the parallelogram is 2. By the same reasoning, the area of the parallelogram determined by And 0 is also 6. Also del [u v] = det And det [u x] = The determinant of the
Construct a random 4 ( 4 matrix A with integer entries between - 9 and 9, and compare det A with det AT, det (-A), det(2A), and det(10 A). Repeat with two other random 4 ( 4 integer matrices, and make conjectures about how these determinants are related. (Refer to Exercise 36 in Section 2.1) Then
Compute the determinants in Exercises 1-2 by cofactor expansions. At each step, choose a row or column that involves the least amount of computation.1.2.
Each equation in Exercise 1-2 illustrates a property of determinants. State the property.1.2.
Combine the methods of row reduction and cofactor expansion to compute the determinists the determines in Exercise 1-2?1.2.
Find the determinants in Exercise 1 - 2 where1. 2.
In Exercise 1-2, use determinants to find out if the matrix is invertible?1.2.
In Exercise 1-2, use determinants to decide if the set of vectors in linearly independent?1.2.
Use Theorem 3 (but not Theorem 4) to show that if two rows of a square matrix A are equal, then det A = 0. The same is true for two columns. Why?
Verify that det A = det B + det C, where
Verify that det A = det B + det C, whereHowever, that A is not the same as B + C?
Right-multiplication by an elementary matrix E affects the columns of A in the same way that left-multiplication affects the rows. Use Theorems 5 and 3 and the obvious fact that ET is another elementary matrix to show that det AE = (det E) (det A?
Compute det ATA and det AAT for several random 4 ( 5 matrices and several random 5 ( 6 matrices. What can you say about ATA and AAT when A has more columns than rows?
If det A is close to zero, is the matrix A nearly singular? Experiment with the nearly singular 4 ( 4 matrix A in Exercise 9 of Section 2.3. Compute the determinants of A, 10A, and 0.1A. In contrast, compute the condition numbers of these matrices. Repeat these calculations when A is the 4 ( 4
Find the determinants in Exercise 1-2 by row reduction to echelon from?1.2.
Use Cramer's rule to compute the solutions of the systems in Exercises 1 - 6? 1. 2.
In Exercises 1-2, compute the adjugate of the given matrix, and then use Theorem 8 to give the inverse of the matrix.1.2.
Show that if A is 2 ( 2, then Theorem 8 gives the same formula for A-1 as that given by Theorem 4 in Section 2.2?
Suppose that all the entries in A are integers and det A = 1. Explain why all the entries in A-1 are integers?
In Exercises 1 - 2, find the area of the parallelogram whose vertices are listed? 1. (0, 0), (5, 2), (6, 4), (11, 6)? 2. (0, 0), (-1, 3), (4, -5), (3, -2)?
Find the volume of the parallelepiped with one vertex at the origin and adjacent vertices at (1, 0, -2), (1, 2, 4), and (7, 1, 0)?
Find the volume of the parallelepiped with one vertex at the origin and adjacent vertices at (1, 4, 0), (-2, -5, 2), and (-1, 2, -1)?
Use the concept of volume to explain way the determinant of a 3 ( 3 matrix A is zero if and only if A is not invertible. Do not appeal to Theorem 4 in Section 3.2. [Think about the columns of A.]
Let T : Rm → Rn be a linear transformation, and let p be a vector and S a set in Rm. Show that the image of p + S under T is the translated set T(p) + T(S) in Rn?
Let S be the parallelogram determined by the vectorsAnd And let Compute the area of the image of S under the mapping x Ax?
Repeat Exercise 27 withAnd
Find a formula for the area of the triangle whose vertices are 0, v1, and v2 in R2?
Let R be the triangle with vertices at (x1, y1), (x2, y2), and (y3, y3). Show thatTranslate R to the origin by subtracting one of the vertices, and use Exercise 29?
Let T : R3 †’ R3 be the linear transformation determined by the matrixWhere a, b, and c are positive numbers. Let S be the unit ball, whose bounding surface has the equation a. Show that T(S) is bounded by the ellipsoid with the equation b. Use the fact that the volume of the unit ball is 4(/3
Let S be the tetrahedron in R3 with vertices at the vectors 0, e1, e2, and e3, and let S' be the tetrahedron with vertices at vectors 0, v1, v2, and v3. See the figurea. Describe a linear transformation that maps S onto S'. b. Find a formula for the volume of the tetrahedron S' using the fact
Test Cramer's rule for a random 4 ( 4 matrix A and a random 4 x 1 vector b. Compute each entry in the solution of Ax = b, and compare these entries with the entries in A-1b. Write the command (or keystrokes) for your matrix program that uses Cramer's rule to produce the second entry of x?
If your version of MATLAB has the flops command, use it to count the number of floating point operations to compute A-1 for a random 30 ( 30 matrix. Compare this number with the number of flops needed to form (adj A) / (det A)?
In Exercises 1-2, determine the values of the parameter s for which the system has a unique solution, and describe the solution.1.2.
Determine the area of the parallelogram determine by the points (1, 4), (-1, 5), (3, 9), and (5, 8). How can you tell that the quadrilateral determined by the points is actually a parallelogram?
Let A, B, C, D, and I be n ( n matrices. Use the definition or properties of a determinant to justify the following formulas. Part (c) is useful in applications of eigenvalues (Chapter 5).a.b.c.
Let A, B, C, and D be n ( n matrices with A invertible.a. Find matrices X and Y to produce the block LU factorizationAnd then show that b. Show that if AC = CA, then
Let J be the n ( n matrix of all 1's and considerA = (a - b)I + b J ; that is,Confirm that det A = (a - b)n-1 [a + (n - 1)b] as follows: a. Subtract row 2 from row 1, row 3 from row 2, and so on, and explain why this does not change the determinant of the matrix. b. With the resulting matrix from
Let A be the original matrix given in Exercise 16, and letNotice that A, B, and C are nearly the same except that the first column of A equals the sum of the first columns of B and C. A linearity property of the determinant function, discussed in Section 3.2, says that det A = det B C det C. Use
Apply the result of Exercise 16 to find the determinants of the following matrices, and confirm your answer using a matrix program?
Use a matrix program to compute the determents of the following matrices?Use the results to guess the determinant of the matrix below, and confirm your guess by using row operations to evaluate that determinant.
Use row operations to show that the determinants in Exercise 1-2 are all zero?1.2.
Use the method of Exercise 19 to guess the determinant ofJustify your conjecture. [Use Exercise 14(c) and the result of Exercise 19]
Compute the determinants in Exercises 1 and 2?1.2.
Show that the equation of the line in R2 through distinct points (x1, y1) and (x2, y2) can be written as
Find a 3 ( 3 determinant equation similar to that in Exercise 7 that describes the equation of the line through (x1, y1) with slope m?
Exercises 9 and 10 concern determinants of the following Vander-monde matrices.1. Use row operations to show that det T = (b - a) (c - a) (c - b) 2. Let f(t) = det V, with x1, x2, x3 all distinct. Explain why f(t) is a cubic polynomial, show that the coefficient of t3 is nonzero, and find three
Let V be the first quadrant in the xy-plane; that is, leta. If u and v are in V, is u + v in V? Why? b. Find a specific vector n in V and a specific scalar c such that cu is not in V. (This is enough to show that V is not a vector space.)
Let H be the set of all vectors of the formWhere t is any real number. Show that h is a subspace of R3. (Use the method of Exercise 9.)
Let W be the set of all vector of the formWhere b and c arbitrary. Find vectors u and v such that W = Span {u, v}. Why does this show that W is a subspace of R3?
Let W be the set of all vectors of the formShow that w is a subspace of R4. (Use the method of Exercise 11.)
LetAnd a. Is w in {v1, v2, v3}? How many vectors are in {v1, v2, v3}? b. How many vectors are in Span {v1, v2, v3}? c. Is w in the subspace spanned by {v1, v2, v3}? Why?
Let v1, v2, v3 be as in Exercise 13, and letIs w in the subspace spanned by {v1, v2, v3}? Why?
In Exercises 1-2, let W be the set of all vectors of the form shown, where a, b, and c represent arbitrary real numbers, In each case, either find a set S of vectors that spans W or give an example to show that W is not a vector space.1.2.
If a mass m is placed at the end of a spring, and if the mass is pulled downward and released, the mass-spring system will begin to oscillate. The displacement y of the mass from its resting position is given by a function of the formWhere ( is a constant that depends on the spring and the mass.
Let W be the union of the first and third quadrants in the xy - plane. That is, leta. If u is an W and c is any scalar, is cu in W? Why?b. Find specific vectors u and v in W such that u + v is not in W. This is enough to show that W is not a vector space?
The set of all continuous real-valued functions defined on a closed interval [a, b] in R is denoted by C[a, b]. This set is a subspace of the vector space of all real-valued functions defined on [a, b], a. What facts about continuous functions should be proved in order to demonstrate that C[a, b]
Determine if the set H of all matrices of the formIs a subspace of M2(2?
Let F be a fixed 3 ( 2 matrix, and let H be the set of all matrices A in M2(4 with property that FA = 0 (the zero matrix in M3(4). Determine if H is a subspace of M2(4?
Prove that (-1)u = -u. (Show that u + (-1) u = 0. Use some axioms and the results of Exercise 1 to 2?
Let H be the set of points inside and on the unit circle in the xy-plane. That is, letFind a specific example - two vectors or vector and a scalar - to show that H is not a subspace of R2?
Suppose cu = 0 for some nonzero scalar c. Show that u = 0. Mention the axioms or properties you use?
Let u and v be vectors in a vector space V, and let H be any subspace of V that contains both u and v. Explain why H also contains Span {u, v}. This shows that Span {u, v} is the smallest subspace of V that contains both u and v?
Let H and K be subspaces of a vector space V. The intersection of H and K, written as H ( K, is the set of v in V that belong to both H and K. Show that H ( K is a subspace of V. (See the figure.) Give an example in R2 to show that the union of two subspaces is not, in general, a subspace.

Showing 3100 - 3200 of 11883

First
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
Last

Step by Step Answers

Services

Sitemap
Fun
Definitions
Become Tutor
Used Textbooks
Study Help Categories
Recent Questions
Expert Questions
Campus Wear
Sell Your Books

Company Info

Security
Copyrights
Privacy Policy
Terms & Conditions
SolutionInn Fee
Scholarship
Online Quiz
Give Feedback, Get Rewards

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Student Discount
Campus Ambassador

Secure Payment

Download Our App

© 2026 SolutionInn. All Rights Reserved