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mathematics
linear algebra and its applications

Linear Algebra And Its Applications 6th Global Edition David Lay, Steven Lay, Judi McDonald - Solutions

Mark each statement True or False (T/F). Justify each answer.(T/F) The equation Ax = 0 gives an explicit description of its solution set.
Mark each statement True or False (T/F). Justify each answer.(T/F) The equation x = x₂u + x3V, with x2 and x3 free (and neither u nor v a multiple of the other), describes a plane through the origin.
Mark each statement True or False (T/F). Justify each answer.(T/F) The homogeneous equation Ax = 0 has the trivial solution if and only if the equation has at least one free variable.
Mark each statement True or False (T/F). Justify each answer.(T/F) The equation Ax = b is homogeneous if the zero vector is a solution.
Mark each statement True or False (T/F). Justify each answer.(T/F) The equation x = p + tv describes a line through v parallel to p.
Mark each statement True or False (T/F). Justify each answer.(T/F) The effect of adding p to a vector is to move the vector in a direction parallel to p.
Mark each statement True or False (T/F). Justify each answer.(T/F) The solution set of Ax = b is the set of all vectors of the form w = p + Vh, where vh is any solution of the equation Ax = 0.
Mark each statement True or False (T/F). Justify each answer.(T/F) The solution set of Ax = b is obtained by translating the solution set of Ax = 0.
Suppose A is the 3 x 3 zero matrix (with all zero entries). Describe the solution set of the equation Ax = 0.
If b ≠ 0, can the solution set of Ax = b be a plane throughthe origin? Explain.
Let A be an m x n matrix, and let u and v be vectors in Rn with the property that Au = 0 and Av = 0. Explain why A(u + v) must be the zero vector. Then explain why A(cu + dv) = 0 for each pair of scalars c and d.
Determine which matrices are in reduced echelon form and which others are only in echelon form.(a)(b)(c)(d) 1 0 0 1 0 0 0 1 1 1 00
Determine which matrices are in reduced echelon form and which others are only in echelon form.(a)(b)(c)(d) 1 0 0 0 0 1 0 01 0 0 1
Row reduce the matrices into reduced echelon form. Circle the pivot positions in the final matrix and in the original matrix, and list the pivot columns. 1 4 6 57 3 68 9
Compute u + v and u – 2v. U= [-2], x= [1 V= -3 3 ]
Compute u + v and u – 2v. = 3 2 V = 2 3
Use the definition of Ax to write the matrix equation as a vector equation, or vice versa. 7 2-9 3 -4 -5 7 -2 [4 6 -9 1 -8 =[2] 44
Write a system of equations that is equivalent to the given vector equation. -8 X₁ -3 + x2 [HH] 7 0 4 2 9 -6 -5
Row reduce the matrices into reduced echelon form. Circle the pivot positions in the final matrix and in the original matrix, and list the pivot columns. 13 3 5 لا لا 357 5 7 9 7 9 1
Use the definition of Ax to write the matrix equation as a vector equation, or vice versa. 7-3 2 1 9-6 2 -3 -5 = 1 -9 12 -4
Describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix. 1 LO 3-3 1 -4 7 5
Consider each matrix as the augmented matrix of a linear system. State in words the next two elementary row operations that should be performed in the process of solving the system. 0 0 1 0 0 1 1 3-4 1 1 5 0 0 0 0 9 -8 7 -6
Use the definition of Ax to write the matrix equation as a vector equation, or vice versa. Xx1 4 -1 7 -4 +x₂ -5 3 -5 + x3 7 -8 0 2 || 6 -8 0
Use the accompanying figure to write each vector listed as a linear combination of u and v. Is every vector in R2 a linear combination of u and v?Vectors a, b, c, and d d -2v b = X 2v y 4 N
Find the general solutions of the systems whose augmented matrices are given. 2 4 7 0 0 7 11
Write a system of equations that is equivalent to the given vector equation. [3] + X1 8 [³] + x₁[_!] = [8] X3 5 -6 +x2
The augmented matrix of a linear system has been reduced by row operations to the form shown. In each case, continue the appropriate row operations and describe the solution set of the original system. 1 7 0 1 -1 0 0 0 0 01-2 3-4 3 1
The augmented matrix of a linear system has been reduced by row operations to the form shown. In each case, continue the appropriate row operations and describe the solution set of the original system. 1 0 0 15 1 9 0 7 0 0 -7
Use the accompanying figure to write each vector listed as a linear combination of u and v. Is every vector in R2 a linear combination of u and v?Vectors w, x, y, and z d -2v b = X 2v y 4 N
Describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix. 1 0 -2 -9 5 12-6
Use the definition of Ax to write the matrix equation as a vector equation, or vice versa. = [ - ] + ² [¯ $ ] + - [¯ ] + - [8] =[J] -2 5 13
The augmented matrix of a linear system has been reduced by row operations to the form shown. In each case, continue the appropriate row operations and describe the solution set of the original system. 1-2 0 1 0 0 0 0 0 3 0-4 1 0 0 1 0 0 0 0
The augmented matrix of a linear system has been reduced by row operations to the form shown. In each case, continue the appropriate row operations and describe the solution set of the original system. 1 -1 0 0-4 1-3 0 -7 -1 4 0 0 0 1 -3 0 0 0 0
Find the general solutions of the systems whose augmented matrices are given. 0 1-2 1-6 5 7 -4
Describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix. 1 2 3 6 0 -4 8- 0
Describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix. 2-8 6 4-3 - 1
Write a vector equation that is equivalent to the given system of equations. X2 + 5x3 = 0 4x1 + 6x2 X3 = 0 -X1 + 3x28x3 8x3 = 0
Determine if b is a linear combination of a1, a2, and a3. 0 ++---------] a2 = аз = 2 1 0 5 8 b = 2 6
Determine if b is a linear combination of a1, a2, and a3. a₁ = 1 -2 2 a₂ = , 0 5 5 , аз 2 -5 = [B]. [] 0 b= 11 8 -7
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. -[₂ A = = 1 0 -2 2 1 -4 4 5 , b = -3 -2 2 9
Find the general solutions of the systems whose augmented matrices are given. 1 -2 -1 3 -6-2 3 2
Solve the systems + 4x3 = -4 + 7x3 = -8 4 x₁3x2 3x17x2 -4x1 + 6x₂ + 2x3:
Describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix. 1 -4-2 0 0 0 0 0 0 1 0 0 0 0 0 0 3-5 0 -1 1 -4 00
Find the general solutions of the systems whose augmented matrices are given. -4 2 3 -9 12 -6 -6 8 -4 0 0 0
Write a vector equation that is equivalent to the given system of equations. 4x1 + x₂ + 3x3 = 9 x17x22x3 = 2 8x16x25x3= 15
Describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix. 1 5 0 0 0 0 000 2-6 1 -7 0 0 0 9 0 4-8 0 1 0 0
LetIs u in the plane in R3spanned by the columns of A? (See the figure.) Why or why not? u= 0 [B]₁ 4 4 and A = 3-5 6 1 -2 1
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. A= 1 2 -3 -1 05 1 2 3 -4 b =
Find the general solutions of the systems whose augmented matrices are given. 1 -7 0 -1 6 0 0 1-2 7-4 2 5 -3 من دل ب 7
Solve the systemsx₂ + 4x3 = -4x1 + 3x2 + 3x3 = -23x1 + 7x2 + 5x3 = 6
Find the general solutions of the systems whose augmented matrices are given. 1-3 0 0 0 0 0 0-1 -1 0 -2 1 9-4 0 0 0 0 0-4 1 0
You may find it helpful to review the information in the Reasonable Answers box from this section before answering.Write down the equations corresponding to the augmented matrix in Exercise 9 and verify your answer to Exercise 9 is correct by substituting the solutions you obtained back into the
LetIs u in the subset of R3 spanned by the columns of A? Why or why not? u= 2 -3 and A = 2 5 0 1 8 1 3 7 -1 0
LetShow that the equation Ax = b does not have a solution for all possible b, and describe the set of all b for which Ax = b does have a solution. 4 = [ 8 ] and b = [b] A 3-4 8 -6
Verify that the solution you found to Exercise 12 is correct by substituting the values you obtained back into the original equations.Data from in Exercise 12Solve the systems + 4x3 = -4 + 7x3 = -8 4 x₁3x2 3x17x2 -4x1 + 6x₂ + 2x3:
You may find it helpful to review the information in the Reasonable Answers box from this section before answering.Write down the equations corresponding to the augmented matrix in Exercise 10 and verify your answer to Exercise 10 is correct by substituting the solutions you obtained back into the
Find the general solutions of the systems whose augmented matrices are given. 1 2 -5 0 0 -4 1 -6 -4 0 0 0 0 0 0 0 0-5 0 2 1 0 0 0
List five vectors in Span {v1, v2}. For each vector, show the weights on v1 and v2 used to generate the vector and list the three entries of the vector. Do not make a sketch. --[-] VI = 1 V2 -6 -5 3 0
List five vectors in Span {v1, v2}. For each vector, show the weights on v1 and v2 used to generate the vector and list the three entries of the vector. Do not make a sketch. V1 = 3 0, V2 2 = 0 3
Verify that the solution you found to Exercise 11 is correct by substituting the values you obtained back into the original equations.Data from in Exercise 11Solve the systemsx₂ + 4x3 = -4x1 + 3x2 + 3x3 = -23x1 + 7x2 + 5x3 = 6
Refer to the matrices A and B below. Make appropriate calculations that justify your answers and mention an appropriate theorem.How many rows of A contain a pivot position? Does the equation Ax = b have a solution for each b in R4? A = 1 -1 0 2 3 0 −1 -1 -4 3 1 2-8 03-1 B = 1 0 1 -2 3-2 2 1
You may find it helpful to review the information in the Reasonable Answers box from this section before answering.Write down the equations corresponding to the augmented matrix in Exercise 11 and verify your answer to Exercise 11 is correct by substituting the solutions you obtained back into the
Verify that the solution you found to Exercise 13 is correct by substituting the values you obtained back into the original equations.Data from in Exercise 13Solve the systems XI - 3x3 = 2x1 + 2x2 + 9x3 = x₂ + 5x3 = 8 7 -2
LetFor what value(s) of h is b in the plane spanned by a1 and a2? a₁ --- 4 a2 -2 -2 -3 7 and b = 4 [1 1 h
LetFor what value(s) of h is y in the plane generated by v1 and v2? VI = 1 -5 []+ [1] 0, V2 = -4 7 , and y = h -5
Refer to the matrices A and B below. Make appropriate calculations that justify your answers and mention an appropriate theorem.Do the columns of B span R4? Does the equation Bx = y have a solution for each y in R4? A = 1 -1 0 2 3 0 −1 -1 -4 3 1 2-8 03-1 B = 1 0 1 -2 3-2 2 1 1-5 2-3 7 2-1 -8
Verify that the solution you found to Exercise 14 is correct by substituting the values you obtained back into the original equations.Data from in Exercise 14Solve the systems X1 - 3x₂ -x₁ + = 5 x₂ + 5x3 = 2 x₂ + x3 = 0
Refer to the matrices A and B below. Make appropriate calculations that justify your answers and mention an appropriate theorem.Can each vector in R4 be written as a linear combination of the columns of the matrix A above? Do the columns of A span R4? A = 1 -1 0 2 3 0 −1 -1 -4 3 1 2-8 03-1 B
Refer to the matrices A and B below. Make appropriate calculations that justify your answers and mention an appropriate theorem.Can every vector in R4 be written as a linear combination of the columns of the matrix B above? Do the columns of B span R3? A = 1 -1 0 2 3 0 −1 -1 -4 3 1 2-8 03-1 B
Determine if the systems are consistent. Do not completely solve the systems. XI 3x1 = 2 X2 - 3х4 = 3 - 2x2 + 3x3 + 2x4 = 1 +7x4 = -5 + 3x3
Use the notation for matrices in echelon form. Suppose each matrix represents the augmented matrix for a system of linear equations. In each case, determine if the system is consistent. If the system is consistent, determine if the solution is unique.(a)(b) 0 0 0 * 0
Use the notation for matrices in echelon form. Suppose each matrix represents the augmented matrix for a system of linear equations. In each case, determine if the system is consistent. If the system is consistent, determine if the solution is unique.(a)(b) 0 0 0 * 0
LetDoes {v1, v2, v3} span R3? Why or why not? -[ 0 0 -2 VI = V2 = 0 38 -3, V3 = 4 -1 -5
Letis in Span {u, v} for all h and k. u= [3] and v= 2 ²]. Show that [*]
You may find it helpful to review the information in the Reasonable Answers box from this section before answering.Write down the equations corresponding to the augmented matrix in Exercise 12 and verify your answer to Exercise 12 is correct by substituting the solutions you obtained back into the
Determine if the systems are consistent. Do not completely solve the systems. X1 2x2 + 2x3 - 2x4 = -3 0 1 5 = X3 + 3x4 = -2x1 + 3x₂ + 2x3 + x4 =
LetDoes {v1, v2, v3} span R4? Why or why not? V1 0 [H] V2 = V₂ 0 0 1 V3 = 1 0 0
Determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system. 2 4 3 لنا 6 h 7
Determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system. 1 -2 h 4 -3 6
Determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system. 6 -4 -3 h -9
Choose h and k such that the system has(a) No solution(b) A unique solution(c) Many solutions. Give separate answers for each part x₁ + 4x₂ = 5 2x1 + hx₂ = k
Determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system. 1 3 -2 h 8 -4
Do the three lines x₁ - 4x2 = 1, 2x₁ - x₂ = −3, and -x1 -3x₂ = 4 have a common point of intersection?Explain.
Determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system. 3-4 8 -6 h 9
Do the three planes x₁ + 2x₂ + x3 = 4, X₂ X3 = 1, and x₁ + 3x₂ = 0 have at least one common point of intersection? Explain.
Construct a 3 x 3 matrix A, with nonzero entries, and a vector b in R3 such that b is not in the set spanned by the columns of A.
Mark each statement True or False (T/F). Justify each answer.(T/F) Any list of five real numbers is a vector in R5.
Mark each statement True or False (T/F). Justify each answer.(T/F) Every matrix equation Ax = b corresponds to a vector equation with the same solution set.
key statements from this section are either quoted directly, restated slightly (but still true), or altered in some way that makes them false in some cases. Mark each statement True or False, and justify your answer. (If true, give the approximate location where a similar statement appears, or
Mark each statement True or False (T/F). Justify each answer.(T/F) If the equation Ax = b is inconsistent, then b is not in the set spanned by the columns of A.
Mark each statement True or False (T/F). Justify each answer.(T/F) The vector u results when a vector u - v is added to the vector v.
Mark each statement True or False (T/F). Justify each answer.(T/F) A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax = b has at least one solution.
Mark each statement True or False (T/F). Justify each answer.(T/F) The echelon form of a matrix is unique.
Mark each statement True or False (T/F). Justify each answer.(T/F) The equation Ax = b is consistent if the augmented matrix [A b] has a pivot position in every row.
Mark each statement True or False (T/F). Justify each answer.(T/F) The weights C₁,..., Cp in a linear combination c₁v₁ +... + CpVp cannot all be zero.
A mining company has two mines. One day’s operation at mine 1 produces ore that contains 20 metric tons of copper and 550 kilograms of silver, while one day’s operation at mine 2 produces ore that contains 30 metric tons of copper and 500 kilograms of silver. LetThen v₁ and v₂ represent the
Mark each statement True or False (T/F). Justify each answer.(T/F) If A is an m × n matrix whose columns do not span Rm, then the equation Ax = b is inconsistent for some b in Rm.
Key statements from this section are either quoted directly, restated slightly (but still true), or altered in some way that makes them false in some cases. Mark each statement True or False, and justify your answer. (If true, give the approximate location where a similar statement appears, or
Mark each statement True or False (T/F). Justify each answer.(T/F) The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process.
Mark each statement True or False (T/F). Justify each answer.(T/F) The solution set of the linear system whose augmented matrix is [a₁ a2 a3 b] is the same as the solution set of the equation x₁ a₁ + x₂a2 + X3a3 = b.

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