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linear algebra
Questions and Answers of
Linear Algebra
Find all the eigenvalues and associated eigenvectors of each of the following matrices:(a)(b) (c) (d)
Prove that if A is an upper (lower) triangular matrix, then the eigenvalues of A are the elements on the main diagonal of A?
Prove that A and AT have the same eigenvalues. What, if anything, can we say about the associated eigenvectors of A and AT?
Let L: V → V be a linear operator, where V is an n-dimensional vector space. Let λ be an eigenvalue of L. Prove that the subset of V consisting of 0V and all eigenvectors of L associated with λ
Let A be an eigenvalue of the n ( n matrix A. Prove that the subset of Rn (Cn) consisting of the zero vector and all eigenvectors of A associated with λ. is a subspace of Rn (Cn). This subspace is
Why do we have to include OV in the set of all eigenvectors associated with A?
In Exercises, find a basis for the eigenspace associated with A for each given matrix.(a)(b)
Let L: P1 → P1 be the linear operator defined by L(at + b) = bt + a. Using the matrix representing L with respect to the basis {1, t} for P1, find the eigenvalues and associated eigenvectors of L?
Prove that if λ is an eigenvalue of a matrix A with associated eigenvector x, and k is a positive integer, then λk is an eigenvalue of the matrix Ak = A ( A ...... A (k factor) With associated
Prove that if Ak = O for some positive integer k then 0 is the only eigenvalue of A?
Let A be an n ( n matrix. (a) Show that det(A) is the product of all the roots of the characteristic polynomial of A. (b) Show that A is singular if and only if 0 is an eigenvalue of A. (c) Also
Let L: V -→ V be an invertible linear operator and let λ be an eigenvalue of L with associated eigenvector x. (a) Show that 1 / λ is an eigenvalue of L-1 with associated eigenvector x? (b) State
Let A be an n ( n matrix with eigenvalues λ1 and λ2, where λ1 ( λ2. Let S1 and S2 be the eigenspaces associated with λ1, and λ2, respectively. Explain why the zero vector is the only vector
Let λ be an eigenvalue of A with associated eigenvector x. Show that λ + r is an eigenvalue of A + rln with associated eigenvector x. Thus, adding a scalar multiple of the identity matrix to A
Let A and B be n ( n matrices such that Ax = λx and Bx = (x. Show that (a) (A + B)x = (λ + ()x? (b) (AB)x = (λ()x?
Let A be an n x n matrix whose characteristic polynomial is p(λ) = λn + a1 λn-1 + .... + an-1 λ + an. If A is nonsingular, show that A-1 = - 1/an (An-1 + a1 An-2 + .... + an-2 A + an-1 In)?
Show that if A is a matrix all of whose columns add up to 1, then λ = 1 is an eigenvalue of A? (Consider the product ATx, where x is a vector all of whose entries are 1, and use Exercise 12.)
Show that if A is an n ( n matrix whose kth row is the same as the kth row of In, then 1 is an eigenvalue of A?
Let A be a square matrix. (a) Suppose that the homogeneous system Ax = 0 has a nontrivial solution x = u. Show that u is an eigenvector of A? (b) Suppose that 0 is an eigenvalue of A and v is an
Find the characteristic polynomial, the eigenvalues, and associated eigenvectors of each of the following matrices:(a)(b) (d) (e)
Find all the eigenvalues and associated eigenvectors of each of the following matrices:(a)(b) (c) (d)
For each of the following matrices find, if possible, a non-singular matrix P such that P-1 A P is diagonal:(a)(b) (c) (d)
Let A be a 2 ( 2 matrix whose eigenvalues are 3 and 4, and associated eigenvectors areAnd Computation, find a diagonal matrix D that is similar to A, and a nonsingular matrix P such that P-1 AP = D?
Which of the following matrices are similar to a diagonal matrix?(a)(b) (c) (d)
Show that none of the following matrices is diagonalizable:(a)(b) (c) (d)
Let L: P1 → P1 be the linear operator defined by L(at + b) = -bt - a. Find, if possible, a basis for P1 with respect to which L is represented by a diagonal matrix?
Let A and B be nonsingular n ( n matrices. Prove that AB and BA have the same eigenvalues?
(Calculus Required) Let V be the vector space of continuous functions with basis {et, e-t}. Let L : V → V be defined by L(g(t)) = g'(t) for g(t) in V. Show that L is diagonalizable?
Prove that if A is diagonalizable, then (a) AT is diagonalizable, and (b) Ak is diagonalizable, where k is a positive integer?
Show that if A is nonsingular and diagonalizable, then A-1 is diagonalizable?
Let λ1, λ2,....., λk be distinct eigenvalues of an n ( n matrix A with associated eigenvectors x1, x2 ..... xk. Prove that x1, x2, ..... , xk are linearly independent?
Show that if a matrix A is similar to a diagonal matrix D, then Tr(A) = Tr(D), where Tr(A) is the trace of A?
(Calculus Required) Let V be the vector space of continuous functions with basis {sint, cos t}, and let L: V → V be defined as L(g(t)) = g'(t). Is L diagonalizable?
Which of the following matrices are diagonalizable?(a)(b) (c) (d)
Find a 2 ( 2 non-diagonal matrix whose eigenvalues are 2 and -3, and associated eigenvectors areAnd Respectively?
`For the orthogonal matrixVerify that (Ax, Ay) = (x, y) for any vectors x and y in R2?
Let A bean n ( n orthogonal matrix, and let L: Rn -→ Rn be the linear operator associated with A; that is, L(x) = Ax for x in Rn. Let 9 be the angle between vectors x and y in R". Prove that the
A linear operator L: V → V, where V is an n-dimensional Euclidean space, is called orthogonal if (L(x), L(y)) = (x, y). Let S be an orthonormal basis for V, and let the matrix A represent the
Let L: R2 → R2 be the linear operator performing a counterclockwise rotation through ( / 4 and let A be the matrix representing L with respect to the natural basis for R2. Prove that A is
Diagonalize each given matrix and find an orthogonal matrix P such that P-l AP is diagonal?1.2.
Diagonalize each given matrix?1.2.
Prove Theorem 7.9 for the 2 ( 2 case by studying the two possible cases for the roots of the characteristic polynomial of A?
Show that if A and B are orthogonal matrices, then AB is an orthogonal matrix?
Let L : V → V be an orthogonal linear operator, where V is an n-dimensional Euclidean space. Show that if λ is an eigenvalue of L, then |λ| = 1?
Let L: R2 R2 be defined byShow that L is an isometry of R2?
Let L: R2 → R2 be defined by L(x) = Ax, for x in R2, where A is an orthogonal matrix? (a) Prove that if det(A) = 1, then L is a counterclockwise rotation? (b) Prove that if det(A) = -1, then L is a
Let L: Rn → Rn be a linear operator? (a) Prove that if L is an isometry, then ||L(x)|| = ||x||, for x in R". (b) Prove that if L is an isometry and ( is the angle between vectors x and y in Rn,
Let L: Rn → Rn be a linear operator defined by L(x) = Ax for x in R". Prove that if L is an isometry, then L-1 is an isometry?
Let L: Rn → Rn be a linear operator and S = {v1, V2,..., v"} an orthonormal basis for Rn. Prove that L is an isometry if and only if T = (L(v1), L(v2),..., L(v")} is an orthonormal basis for Rn?
Show that if A is an orthogonal matrix, then A-1 is orthogonal?
(a) Verify that the matrixIs orthogonal.(b) Prove that if A is an orthogonal 2 ( 2 matrix, then there exists a real number ( such that eitherOr
Let A be any n ( n real matrix.(a) Prove that the coefficient of (n-1 in the characteristic polynomial of A is given by - Tr(A).(b) Prove that Tr(A) is the sum of the eigenvalues of A.(c) Prove that
Prove or disprove: Every nonsingular matrix is similar to a diagonal matrix?
Let P(x) = a0 + a1x + a2x2 + ... + akxk be a polynomial is x. Show that the eigenvalues of matrix p(A) = a0In + a1A + a2A2 + .... + akAk are p(λi), i = 1, 2, ......, n, where λi are the eignevalues
Let V = M22 and let L : V V be the linear operator defined by L(A) = AT, for A in V. Let S = {A1, A2, A3, A4}, where(a) Find [L(Ai)]s for i = 1,2,3,4. (b) Find the matrix B representing
Psychologist: places a rat each day in a cage with two doors, A and B. The rat can go through door A, where it receives an electric shock, or through door B, where it receives some food. A record is
Consider the transition matrix(a) If Compute x(1), x(2), x(3), and x(4) to three decimal places. (b) Show that T is regular and find its steady state vector?
Show that each of the following transition matrices reaches a state of equilibrium?(a)(b) (c) (d)
Determine the singular value decomposition of
Determine the singular value decomposition of
An m ( n matrix A is said to have full rank if rank A = minimum [m, n]. The singular value decomposition lets us measure how close A is to not having full rank. If any singular value is zero, then A
Prove Corollary 8.1, using Theorem 8.1 and the singular value decomposition?
For the five most populated cities in the United States in 2002, we have the following crime information: For property crimes known to police per 100,000 residents, the number of burglaries appears
For the data in Exercise 13, determine the first principal component?
In Section 5.3 we defined a positive definite matrix as a square symmetric matrix C such that yTCy > 0 for every nonzero vector y in Rn. Prove that any eigenvalue of a positive definite matrix is
Let Sn be a covariance matrix satisfying the hypotheses of Theorem 8.6. To prove Theorem 8.7, proceed as follows: (a) Show that the trace of Sn is the total variance. (See Section 1.3, Exercise 43,
Find the dominant eigenvalue of each of the following matrices:(a)(b)
Suppose that we have a system consisting of two interconnected tanks, each containing a brine solution. Tank A contains x(t) pounds of salt in 200 gallons of brine, and tank B contains y(t) pounds of
Consider the linear system of differential equations(a) Find the general solution. (b) Find the solution to the initial value problem determined by the initial conditions X1(0) = 2, x2(0) = 7, x3 (0)
Prove that the set of all solutions to the homogeneous linear system of differential equations x' = Ax, where A is n ( n, is a subspace of the vector space of all differentiable real-valued n-vector
For each of the dynamical systems in Exercises 1 through 3, determine the nature of the equilibrium point at the origin, and describe the phase portrait.(1)(2)(3)
In Exercise 1 write each quadratic form as xT Ax. Where A is a symmetric matrix?(a)(b) (c)
In Exercises 1, which of the given quadratic forms in three variables are equivalent?
If A, B, and C are n ( n symmetric matrices, prove the following: (a) A is congruent to A? (b) If B is congruent to A, then A is congruent to B? (c) If C is congruent to B and B is congruent to A,
Prove that a symmetric matrix A is positive definite if and only if A = PTP for a nonsingular matrix P?
In Exercises 1, for each given symmetric matrix A, find a diagonal matrix D that is congruent to A?(a)(b) (c)
In Exercise 11 through 18, translate axes to identify the graph of each equation and write each equation in standard form? 1. x2 - y2 + 4x - 6y - 9 = 0 2. x2 - 4x + 4y + 4 = 0 3. 4x2 + 5y2 - 30y + 25
In Exercise 19 through 24, rotate exes to identify the graph of each equation and write each equation in standard form? 1. xy = 1 2. x2 + y2 + 4xy = 9 3. 9x2 + 6y2 + 4xy - 5 = 0
In exercise 25 through 30, identify the graph and each equation and write each equation in standard form?1. 5x2 + 5y2 - 6xy - 30(2x + 18(2y + 82 = 02. 6x2 + 9y2 - 4xy - 4(5x - 18(5 y = 53. 8x2 + 8y2
In exercise 15 through 28, classify the quadric surface given by each equation and determine its standard form? 1. -x2 -y2 - z2 + 4xy + 4xz + 4yz = 3 2. 4x2 + 4y2 + 8z2 + 4xy - 4xz - 4yz + 6x - 10y +
During a local campaign, eight Republican and five Democratic candidates are nominated for president of the school board. (a) If the president is to be one of these candidates, how many possibilities
Pamela has 15 different books. In how many ways can she place her books on two shelves so that there is at least one book on each shelf? (Consider the books in each arrangement to be stacked one next
Three small towns, designated by A, B, and C, are interconnected by a system of two-way roads, as shown in Fig. 1.4.(a) In how many ways can Linda travel from town A to town C? (b) How many different
List all the permutations for the letters a, c, t.
Evaluate each of the following. (a) P(7, 2) (b) P(8, 4) (c) P(10, 7) (d) P(12, 3)
An alphabet of 40 symbols is used for transmitting messages in a communication system. How many distinct messages (lists of symbols) of 25 symbols can the transmitter generate if symbols can be
In the Internet each network interface of a computer is assigned one, or more, Internet addresses. The nature of these Internet addresses is dependent on network size. For the Internet Standard
A computer science professor has seven different programming books on a bookshelf. Three of the books deal with C++, the other four with Java. In how many ways can the professor arrange these books
Answer part (c) of Example 1.6. If repetitions are allowed, as in part (b), how many of the plates have only vowels (A, E, I, O, U) and even digits? (0 is an even integer.)
(a) How many arrangements are there of all the letters in SOCIOLOGICAL? (b) In how many of the arrangements in part (a) are A and G adjacent? (c) In how many of the arrangements in part (a) are all
How many positive integers n can we form using the digits 3, 4, 4, 5, 5, 6, 7 if we want n to exceed 5,000,000?
Twelve clay targets (identical in shape) are arranged in four hanging columns, as shown in Fig. 1.5. There are four red targets in the first column, three white ones in the second column, two green
Show that for all integers n, r ¥ 0, if w + 1 > r, then
Find the value(s) of n in each of the following: (a) P(n, 2) = 90, (b) P(n, 3) = 3P(n, 2), and (c) 2P(n, 2) + 50 = P(2n, 2)
How many different paths in the xy-plane are there from (0, 0) to (7, 7) if a path proceeds one step at a time by going either one space to the right (R) or one space upward (U)? How many such paths
(a) How many distinct paths are there from (- 1, 2, 0) to (1, 3, 7) in Euclidean three-space if each move is one of the following types? (H): (x, y, z) → (x + 1, y, z); (V): (x, y, z) → (x, y +
(a) Determine the value of the integer variable counter after execution of the following program segment. (Here i, j, and k are integer variables.) counter : = 0 for i := 1 to 12 do counter : =
Buick automobiles come in four models, 12 colors, three engine sizes, and two transmission types, (a) How many distinct Buicks can be manufactured? (b) If one of the available colors is blue, how
A sequence of letters of the form abcba, where the expression is unchanged upon reversing order, is an example of a palindrome (of five letters), (a) If a letter may appear more than twice, how many
Determine the number of six-digit integers (no leading zeros) in which (a) No digit may be repeated; (b) Digits may be repeated. Answer parts (a) and (b) with the extra condition that the six-digit
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