Questions and Answers of Linear Algebra

Compute the determinants using cofactor expansion along any row or column that seems convenient. al |0 d e h i j
Compute the determinants using cofactor expansion along any row or column that seems convenient. 3 -2 0 -1 2 2 4 3.
Compute the determinants using cofactor expansion along any row or column that seems convenient. -1 0 3 5 2 4 2
Compute the determinants using cofactor expansion along any row or column that seems convenient. sin 0 sin 0 cos 0 tan 0 cos 0 - cos 0 sin 0
Compute the determinants using cofactor expansion along any row or column that seems convenient. 3 -4 4 -3 -1 -2
Compute the determinants using cofactor expansion along the first row and along the first column. 2 3 5 6 8 9
Compute the determinants using cofactor expansion along the first row and along the first column. 0|
Compute the determinants using cofactor expansion along the first row and along the first column. -2 3 3 -1
(a) Show that the eigenvalues of the 2 Ã— 2 matrixare the solutions of the quadratic equation Î»2 - tr(A)Î» + det A = 0, where tr(A) is the trace of A. (See page
Find all of the eigenvalues of the matrix A over the complex numbers C. Give bases for each of the corresponding eigenspaces. 1- 2i 4 A = 1 + 2i
Find all of the eigenvalues of the matrix A over the complex numbers C. Give bases for each of the corresponding eigenspaces. A = [i
Find all of the eigenvalues of the matrix A over the complex numbers C. Give bases for each of the corresponding eigenspaces. -2 A = 1
Use the method of Example 4.5 to find all of the eigenvalues of the matrix A. Give bases for each of the corresponding eigenspaces. Illustrate the eigenspaces and the effect of multiplying
Use the method of Example 4.5 to find all of the eigenvalues of the matrix A. Give bases for each of the corresponding eigenspaces. Illustrate the eigenspaces and the effect of multiplying
Use the method of Example 4.5 to find all of the eigenvalues of the matrix A. Give bases for each of the corresponding eigenspaces. Illustrate the eigenspaces and the effect of multiplying
The unit vectors x in R2and their images Ax under the action of a 2 Ã— 2 matrix A are drawn head-totail, as in Figure 4.7. Estimate the eigenvectors and eigenvalues of A from each
Find the eigenvalues and eigenvectors of A geometrically.(counterclockwise rotation of 90° about the origin) -1 A = %3D 0_
Find the eigenvalues and eigenvectors of A geometrically.(Stretching by a factor of 2 horizontally and a factor of 3 vertically) 2 0 A =
Find the eigenvalues and eigenvectors of A geometrically.(projection onto the line through the origin with direction vector A = || 3 L5
Show that l is an eigenvalue of A and find one eigenvector corresponding to this eigenvalue.Discuss. -1 1 1,A = -1 2 0 1
Show that l is an eigenvalue of A and find one eigenvector corresponding to this eigenvalue. -1 1 1,A = -1 2 0 1
Show that l is an eigenvalue of A and find one eigenvector corresponding to this eigenvalue. 2 A = 2 = -2 -1
Show that v is an eigenvector of A and find the corresponding eigenvalue. -2 3 -1 |A = |, v =
Show that v is an eigenvector of A and find the corresponding eigenvalue. 4 A = [5 -2 -7 10
Show that v is an eigenvector of A and find the corresponding eigenvalue. 2 3 , V 3 2 A = -2
Find an LU factorization of A = | 3 -1 1
If A is a matrix such thatfind A. 1/2 -1 -3/2 4-
Let and Compute the indicated matrices, if possible.ATB2 2 A = 3 5 . -1 2 B = 3 -3
Let and Compute the indicated matrices, if possible.A2B 2 A = 3 5 . -1 2 B = 3 -3
In Theorem 3.37, prove that if B = A, then PGx = 0 for every x in Zk2. (Throughout this proof we denote by ai the ith column of a matrix A.) With P and G as in the statement of the theorem, assume
Construct standard parity check and generator matrices for a (15, 11) Hamming code.
If A is a square matrix whose rows add up to the zero vector, explain why A cannot be invertible.
Ifand X is a matrix such thatfind X. -1 A = | 2 3 -1 L0 1. -1 -3 AX = 5 3 -2
Let and Compute the indicated matrices, if possible.The outer product expansion of AAT 2 A = 3 5 . -1 2 B = 3 -3
Let and Compute the indicated matrices, if possible.(BTB)-1 2 A = 3 5 . -1 2 B = 3 -3
Let and Compute the indicated matrices, if possible.(BBT)-1 2 A = 3 5 . -1 2 B = 3 -3
Let and Compute the indicated matrices, if possible.BTA-1B 2 A = 3 5 . -1 2 B = 3 -3
Mark each of the following statements true or false:(a) For any matrix A, both AAT and ATA are defined.(b) If A and B are matrices such that AB = O and A ≠ O, then B = O.(c) If
In Theorem 3.37, prove that if pi= pj, then we cannot determine whether an error occurs in the ith or the jth component of the received vector. (Throughout this proof we denote by ai the ith column
Show that the code in Example 3.70 is a (3, 1) Hamming code.
Define a code Z32†’ Z62 using the standard generator matrix(a) List all eight code words.(b) Find the associated standard parity check matrix for this code. Is this code (single)
Define a code Z22†’ Z52using the standard generator matrix(a) List all four code words.(b) Find the associated standard parity check matrix for this code. Is this code (single)
The parity check code in Example 1.37 is a code Z62 → Z72.(a) Find a standard parity check matrix for this code.(b) Find a standard generator matrix.
When the (7,4) Hamming code of Example 3.71 is used, suppose the messages c′ are received. Apply the standard parity check matrix to c′ to determine whether an error has occurred and correctly
When the (7,4) Hamming code of Example 3.71 is used, suppose the messages c′ are received. Apply the standard parity check matrix to c′ to determine whether an error has occurred and correctly
When the (7,4) Hamming code of Example 3.71 is used, suppose the messages c′ are received. Apply the standard parity check matrix to c′ to determine whether an error has occurred and correctly
What is the result of encoding the messages using the (7, 4) Hamming code of Example 3.71? х %3 х
What is the result of encoding the messages using the (7, 4) Hamming code of Example 3.71? х—
What is the result of encoding the messages using the (7, 4) Hamming code of Example 3.71? х%3 х 3
Suppose we encode the binary digits 0 and 1 by repeating each digit five times. Thus,0→ [0,0,0,0,0]1→ [1,1,1,1,1]Show that this code can correct double errors.
Suppose we encode the four vectors in by repeating the vector twice. Thus, we have[0,0]→ [0,0,0,0][0,1]→ [0,1,0,1][1,0]→ [1,0,1,0][1,1]→ [1,1,1,1]Show that this code is not error-correcting.
Use powers of adjacency matrices to determine the number of paths of the specified length between the given vertices.Exercise 58, length 3, v1 to v4Data From Exercise 58 1 0 10
Use powers of adjacency matrices to determine the number of paths of the specified length between the given vertices.Exercise 58, length 3, v4 to v1Data From Exercise 58 1 0 10
Use powers of adjacency matrices to determine the number of paths of the specified length between the given vertices.Exercise 55, length 3, v4 to v1Data From Exercise 55 0 1 0 0
Use powers of adjacency matrices to determine the number of paths of the specified length between the given vertices.Exercise 55, length 2, v1 to v3Data From Exercise 55 0 1 0 0
Draw a digraph that has the given adjacency matrix. 0 1 0 0 1 0 1 0 _1
Determine the adjacency matrix of the given digraph. V2 V1 V5 V3 V4
Draw a graph that has the given adjacency matrix. 0 0 1 0 0 1 1 1 0 0 1 1 1 1 1 0 0 1 1 1 0 0
Determine the adjacency matrix of the given graph. V3 V2 V1 V5 V4
Let A be an n × n matrix, A ≥ O. Suppose that Ax < x for some x in Rn, x ≥ 0. Prove that x > 0.
A consumption matrix C and a demand vector d are given. In each case, find a feasible production vector x that satisfies equation (2). 0.1 0.4 1.1 0.1 1.1 C = | 0 0.2 0.2 , d 3.5 0.3 0.2 0.3 2.0
A consumption matrix C and a demand vector d are given. In each case, find a feasible production vector x that satisfies equation (2). 0.5 0.2 0.1 3 0.2 , d = C = | 0 0.4 2 0.5
Determine whether the given consumption matrix is productive. 0.2 0.4 0.1 0.4 0.3 0.2 0.2 0.1 0.4 0.5 0.3 0.5 0 0.2 0.2
Determine which of the matrices are exchange matrices. For those that are exchange matrices, find a non-negative price vector that satisfies equation (1). 0.50 0.70 0.35 0.25 0.30 0.25 0.25 0.40
Determine which of the matrices are exchange matrices. For those that are exchange matrices, find a non-negative price vector that satisfies equation (1). 0.3 0 0.2 0.3 0.5 0.3 0.4 0.5 0.5
Determine which of the matrices are exchange matrices. For those that are exchange matrices, find a non-negative price vector that satisfies equation (1). 1/2 0 1/3 [1/2 0 2/3.
Determine which of the matrices are exchange matrices. For those that are exchange matrices, find a non-negative price vector that satisfies equation (1). [ 1/3 3/2 , 1/3 1/3 -1/2
Determine which of the matrices are exchange matrices. For those that are exchange matrices, find a non-negative price vector that satisfies equation (1). 0.1 0.6 0.4 0.9
Let be the transition matrix for a Markov chain with three states.Letbe the initial state vector for the population.Find the steady state vector. P = 1/3 || 120 X, 180 90
Let be the transition matrix for a Markov chain with three states.Letbe the initial state vector for the population.What proportion of the state 2 population will be in state 3 after two steps?
Let be the transition matrix for a Markov chain with three states.Letbe the initial state vector for the population.What proportion of the state 1 population will be in state 1 after two steps?
Let be the transition matrix for a Markov chain with three states.Letbe the initial state vector for the population.Compute x1 and x2. P = 1/3 || 120 X, 180 90
Letbe the transition matrix for a Markov chain with two states.Letbe the initial state vector for the population.Find the steady state vector. 0.5 0.3 P = 0.5 0.7. 0.5 Xo 0.5
Letbe the transition matrix for a Markov chain with two states.Letbe the initial state vector for the population.What proportion of the state 2 population will be in state 2 after two steps? 0.5 0.3
Letbe the transition matrix for a Markov chain with two states.Letbe the initial state vector for the population.What proportion of the state 1 population will be in state 2 after two steps? 0.5 0.3
Letbe the transition matrix for a Markov chain with two states.Letbe the initial state vector for the population.Compute x1 and x2. 0.5 0.3 P = 0.5 0.7. 0.5 Xo 0.5
Let ABCD be the square with vertices (- 1, 1), (1, 1), (1, - 1), and (- l, - 1). Find and draw the image of ABCD under the given transformation.T in Exercise 31Data From Exercise 31 х + 2х, —
Let ABCD be the square with vertices (- 1, 1), (1, 1), (1, - 1), and (- l, - 1). Find and draw the image of ABCD under the given transformation.The projection in Exercise 22Data From Exercise
Use matrices to prove the given statements about transformations from R2 to R2. If ℓ, m, and n are three lines through the origin, then Fn o Fm o Fℓ is also a reflection in a line through
Use matrices to prove the given statements about transformations from R2 to R2.(a) If P is a projection, then P ο P P.(b) The matrix of a projection can never be invertible.
Find the standard matrix of the composite transformation from R2 to R2.Reflection in the line y = x, followed by counterclockwise rotation through 30°, followed by reflection in the line y = -x
Find the standard matrix of the composite transformation from R2 to R2.Clockwise rotation through 45°, followed by projection onto the y-axis, followed by clockwise rotation through 45°
Verify Theorem 3.32 by finding the matrix of S º T(a) By direct substitution(b) By matrix multiplication of [S][T]. У — У2 X + х X1 У1 X, + хз , . S y2 У2 — Уз |T х — Ул +
Verify Theorem 3.32 by finding the matrix of S º T(a) By direct substitution(b) By matrix multiplication of [S][T]. X1 У1 — У2 х + 2х, X3. У1 T х2 Ул + у2 - У + у2- [ 2х, Xз
Verify Theorem 3.32 by finding the matrix of S º T(a) By direct substitution(b) By matrix multiplication of [S][T]. X1 [ 4y1 – 2y2 — У + Ут х, + х, — х, T X2 2х — х, + хз. У2
Verify Theorem 3.32 by finding the matrix of S —¦ T (a) by direct substitution and (b) by matrix multiplication of [S][T]. Ул + Зу, --A)- X1 X2 2yı + y2 .У». . У1 — У2
Solve the system Ax = b using the given LU factorization of A. 4 3 -2 2 -5 -1 -2 6. 3 -3 -4 3 -2 -5 9_ 4 -2 1_ -8 3 -3 3 b = -2 -1
Find the standard matrix of the given linear transformation from R2 to R2.Reflection in the line y = -x
Find the standard matrix of the given linear transformation from R2 to R2.Reflection in the line y = x
Find the standard matrix of the given linear transformation from R2 to R2.Projection onto the line y = -x
Show that the given transformation from R2 to R2 is linear by showing that it is a matrix transformation.P projects a vector onto the line y = x.
Find the standard matrix of the linear transformation in the given exercise.Data From Exercise 6 х х х+у T\ y х+ у+z.
Find the standard matrix of the linear transformation in the given exercise.Data From Exercise 5 х х — у +z 2х + у — 32]
Find the standard matrix of the linear transformation in the given exercise.Data From Exercise 4 х+ 2у х — х У 3х — 7у.
Give a counterexample to show that the given transformation is not a linear transformation. х+1 х .У,
Prove that the given transformation is a linear transformation, using the definition (or the Remark following Example 3.55). х х х+у T| y х+у+z
Prove that the given transformation is a linear transformation, using the definition (or the Remark following Example 3.55). х х — у+ z 2х + у — 32] T y - ||
Prove that the given transformation is a linear transformation, using the definition (or the Remark following Example 3.55). х+ 2у х — х У Зх — 7у]
compute the rank and nullity of the given matrices over the indicated Zp.over Z7 2 4 5 1 0 0 1 6 3 2 2 5 1 1 11
Doform a basis for Z33? 1 1 1

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