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Differential Equations and Linear Algebra 2nd edition Jerry Farlow, James E. Hall, Jean Marie McDill, Beverly H. West - Solutions

Solutions Ip Tandem: A student who was asked to solve two systems.Noticed that they differed only in their right-hand sides She formed the matrix And obtained its RREF using Gauss-Jordan elimination: She concluded that the solutions of the two systems were given. respectively, by the
Tandem with a Twist(a) Use the method of the previous problem to solve the systems(b) Explain how the calculation of part (a) can be used to solve the matrix equation For the unknown 3 ( 2 matrix X.
Two-Thousand-Year-Old Problem Find the area of two fields, given that one field yields 2/3 of a bushel of wheat per square yard and the other yields 1/2 a bushel per square0 yard. The total area of the two fields is 1,800 square yards and the total yield is 1, 100 bushels. (This is a typical
Computerizing list (in appropriate order) the operations you would need to use to instruct a computer to solve Ax̅ = b̅ by Gauss Jordan elimination: 1. In the 2 ( 2 case, 2. In the 3 ( 3 case The strategy used in Example 3 is a good start for Problems 74 and 75, but it needs to be refined to meet
More Circuit Analysis: For each circuit in Problems 1-3 use Kirchoff's current law (Problem 76) to write a system of equations that must be satisfied by the currents.1.2. 3.
Checking Inverses In each of Problems 1-2, verify by multiplication that the given pair of matrices are inverses of one another.1.2.
Inverse of the 2 ( 2 Matrix: Verify that the inverse of a square matrix A of order two is given by the following handy formula, if ad - bc ( 0. That is, show that AA-1 = I.
Brute Force Compute the inverse of matrixBy setting And solving a system o f equations for a . b. c , and d?
Finding Counterexample For problem 1 and 2 answer the given question by finding a proof or a counterexample in M22. 1. Is it true that the sum of invertible n ( n matrices is invertible? 2. Is it true that the only n ( n matrices for which A2 = A are the zero matrix and the identify matrix?
Unique Inverse Show that if a matrix has an inverse it is unique?
Invertible Matrix Method Use the inverse rnatrix found in Problem 6 to solve the system x1 + 3x2 = -4, 2x1 + 5x2 = 10.
Solution b y Invertible Matrix Use the inverse matrix determined in Problem 7 to solve the system y + z = 5, 5x + v - z = 2. 3x - 3J - 3z = 0.
More Solutions by Invertible Matrices Use the inverse of the coefficient matrix to solve the systems in Problems 1 and 2. 1. x - y + z = 4 x + y = 1 x + 2y - z = 0 2. 4x + 3y - 2z = 0 5x + 6y + 3z = 10 3x + 5y + 2z = 2
Noninvertible 2 ( 2 Matrices Prove that forA is not invertible if ad = be?
Matrix Algebra with Inverses In Problems 1-3. Assume that A and B are invertible matrices of the same order. 1. Simplify (AB-1)-1/ 2. Simplify B(A2B2)-1. 3. If A(BA)-1 x̅ = b̅, solve for x̅,
Question of invertibility: What condition is required in order to solve for i when (A + B)x̅ = b̅?
1. Cancellation Works Prove using matrix algebra that if A, 8, and C are matrices such that AB = AC, and A is invertible, then B = C? 2. An Inverse Prove that if A is invertible and B is another square matrix such that AB = I, then B = A-1?
Making Invertible Matrices Choose a constant k so that the matrices given in Problems 1 and 2 are invertible. If no such k exists, say so and explain your reasoning.1.2.
Products and Noninvertibility Let A and B be n ( n matrices? (a) Show that if BA = In, then AB = I,. . (b) Show that if AB is invertible, then A must be invertible.
Invertibility of Diagonal Matrices Show that a diagonal matrix A is invertible if and only if all diagonal elements are nonzero. Give the form of A-l?
lnvertibility of Triangular Matrices Show that an upper triangular matrix is invertible if and only if all diagonal elements are nonzero?
Inconsistency If the matrix-vector equation Ax̄ = b̅ is inconsistent for some b̅ in R", what can you determine about matrix A?
Inverse of an Inverse Prove the following property (stated in this section): "If A is invertible, then so is A-1 • and (A-1) -1 = A."
Inverse of a Transpose Prove the following property (stated in this section): "If A is invertible, then so is AT and (AT)-1 = (A-l)T."?
Elementary Matrices If we perform a single row operation on an identity matrices, we obtain an elementary matrix E1nt· ERcph or Escale· Find the elementary matrices for each of the following row operations on I3. (a) Interchange rows 1 and 2 (Eint). (b) Add k limes row 1 lo row 3 (ERepl). (c)
lnvertibllity of Elementary Matrices Explain why all elementary matrices must be invertible. Demonstrate this property b y finding the inverses of Eint, ERepl, and Escale?
Similar Matrices Prove the statements in Problems 1-3 given the following definition? A matrix B is defined to be similar to matrix A (denoted by B ~ A) if there is an invertible matrix P such that B = P-1 AP? 1. Matrix B is similar to itself; that is, B ~ B. 2. If B ~ A, then A ~ B. 3. lf A ~ B
If B = p-1 AP for some invertible P, then Bn = p-1 AnP for any positive integer n?
True / False Questions If true, explain. Iffalse, give a 2 ( 2 counterexample. (a) A diagonal matrix is invertible if and only if its diagonal elements are nonzero. True or false? (b) An upper triangular matrix is invertible if and only if its diagonal elements are nonzero. True or false? (c) If A
Leontief Model Find the total outputs for each input output matrix in Problems 1-3, with demands as given?1.2. 3.
Matrix Inverses. Use mw reduction to calculate the inverse of each matrix in Problems 1-3. Consider k ( 0.1.2. 3.
How Much Is Left Over? In Example 5, suppose that A produces $150 worth of its product and B produces $250 worth of its product. What is the dollar value of both products available for external consumption?
Israeli Economy In 1 966, Leontief used his input-output model to analyze the Israeli economy by dividing it into three segments: agriculture (A), manufacturing (M), and energy (E), as shown in the following technological matrix.The export demands on the Israeli economy (in thousands of Israeli
Calculating Determinants Use cofactor expansion and/or row reduction to evaluate the determinant of each matrix in Problems 1-3. (Choose your row or column carefully.)1.2. 3.
1. Check 1he result for | A | using the cofactor expansion. 2. Use the basketweave method to find the determinant for the matrix. in Problem I. 3. Use the basketweave method to find the determinant for the matrix in Problem 2.
Show that the basketweave method does not generalize to finding determinants for higher-order matrices. Try a generalized basketweave method onAnd show that it does not match the correct answer. Which can be obtained by the cofactor method?
Triangular Determinants Show that the determinant of an upper triangular matrix is the product of its diagonal elements. The diagonal matrix is a special case. The statement for a lower triangular matrix can be proved in a similar fashion. For each cofactor expansion. use the first column?
Think Diagonal. Use the ideas of Problem 15 to evaluate the determinants in Problems 1-3?1.2. 3.
lnvertibility In Problems 1-3, What choices of k and m would make the matrices invertible? Check the determinant.1.2. 3. Invertibility Test: In Problems 1-3, use the determinations of the matrices to test for the invertibility of the matrices.
Product Verification For the given 2 ( 2 matrices A and B, show directly (by finding the product AB) that |AB| = |A| |B|1.2.
1. Determinant of an Inverse Prove for invertible matrix A that2. Do Determinants Commute? Let A and B be any two n ( n matrices. Explain why | AB | = | BA |.
Determinant of Similar Matrices Matrix A is said to be similar to matrix B if there is an invertible matrix P such that A = p-1 BP. (This will be discussed in Sec. 5.4 on diagonalization.) Show that similar matrices A and B have the same determinant?
Determinant of A" The notation A" denotes(a) If |An | = 0 for some integer n, then A must be noninvertible. Show why this result is true. (b) If |An | ( 0 for some integer n, then A must be invertible. Verify this result for n = 4?
Determinants of Sums Give an example of square matrices A and B for which |A + B| ( |A| + |B|?
Determinants of Sums Again Give an example of nonzero square matrices A and B such that |A + B| = |A| + |B|?
Scalar Multiplication Determine |kA| in terms of k and |A|?
Inversion by Determinants Let A be a square matrix and let A be its cofactor matrix, the matrix obtained from A by replacing each of its elements by its cofactor. Then if |A| ( 0.Use this formula to compute the inverse of the matrixCheck your result by computing the inverse using row reduction?
Determinants of Elementary Matrices Find the determinants for each of the elementary matrices (Sec. 3.3, Problems 40 and 41) fanned by the elementary row operations on the 3 x 3 identity matrix I, described as follows. (a) Interchange two rows of I to get matrix Eint ( Find |Eint|. (b) Replace a
Determinant of a Product Complete the proof that the determinant of a product is the product of the determinants, using the results of the previous problem and, from Sec. 3.3, Problems 40 and 41. (a) Show that if A is not invertible, then |AB| = |A| | B |. By Problem 34 in Sec. 3.3, if A is not
Cramer's Rule Solve each of the systems in problem 1-3 by employing Cramer's Rule. In problem 2, ( is a parameter? 1. x + 2y = 2 2x + 5y = 0 2. x + y = ( x + 2y = 1 3. x + y + 3z = 5 2y + 5z = 7 x + 2z = 3
The Wheatstone Bridge The Wheatstone bridge is a device used to measure an unknown resistance RA by comparing it with known resistances R1, R2, and R3.5 The circuit diagram is shown in Fig. 3.4.3. The know resistances are adjusted so that no current from the cell with voltage E0 passes through the
Least Squares Derivation Derive equations (8) in the text for the parameters k and m in the least squares equation y = k + mx by simplifying and solving the equations (F / (k = 0, (F /(m = 0, where F is as given in equation (7)?
Alternative Derivation of Least Squares Equations Let(a) Show that equation (9) has matrix vector form AxÌ… = bÌ…. (b) Show that premultiplying each side of the equation in part (a) by AT leads to the least squares equations (8) for n = 4.
Least Squares Calculation Obtain the least squares approximation k + mx for the data that follows. Plot the points and the line
Compute or Calculate For the data sets in Problems 1 and 2, set up the system of equations to find a least squares fit. (A spreadsheet is a very fast way to compute the sums, and to make a scatter plot.) Then solve the system to find the actual least squares line y = k + mx. Finally, plot points
Least Squares in Another Dimension A chemist wishes to estimate the yield Y in a certain process that depends on the temperature T and the pressure P. A linear model for the process is assumed:Y = a + b1T + b2P.Where a, b1, and b2 are unknown parameters. Assume that the observations are as
Least Square System Solution The over determined system of equation AxÌ… = bÌ… given byHas no solution. If you premultiply each side of the equation by AT, however, you will obtain a system with a unique solution xÌ… = [x1. x2] representing a point that minimize
Find the Properties State which one of the row operations for determinants is illustrated in each of Problems 1-3.1.2. 3. Basketweave for 3 ( 3 There is a shortcut for finding the determinant of a 3 ( 3 matrix. For example. let And repeat the first and second columns to the right of the original
They Do Not All Look Like Vectors for each vector space in Problems 1-2 give several examples of vector in the space, including the zero vector and the negative of a typical vector. 1. R2 2. R3 3. R4
Are They Vector Spaces? In each of Problems 1-3, decide whether or not the given set constitutes a vector space. Assume "standard" definitions of the operations? 1. The set of vectors in the first quadrant of the plane? 2. The set of vectors in the first octant of (x, y, z) space? 3. The set of all
A Familiar Vector Space Show that the set R of real numbers is itself a vector space. (We could write the name of this vector space as R1)?
1. Not a Vector Space: show that the set Z of integers (standard operations) is not a vector space by identifying at least one vector space property that fails? 2. DE Solution Space: Verify or justify with results from algebra or calculus the vector space properties 3, 4, and 1-2 of a vector space
Another Solution Space Verify or justify with results from algebra or calculus the vector space properties 3, 4, and 1-2 of a vector space for the solution space of Example 4?
The Space C(-(,() Verify or justify with results from algebra or calculus the fact that the continuous functions on the real line form a vector space?
Vector Space Properties: Show that the properties in Problems 1-3 hold in any vector space? 1. Unique Zero: The zero element in a vector space is unique. Start with two zero elements and show that they must be equal. 2. Unique Negative: The negative of a vector is unique? 3. Zero as Multiplier: For
A Vector Space Equation Suppose that, for c E R and in vector space v̅ in vector space V, cv̅ = 0̅. Then show that either c = 0 or v̅ = 0̅?
Nonstandard Definitions: In Problems 1-2 we explore the possibility of defining a new vector space whose vectors are the familiar pairs (x, y) of real numbers but for which different operations are defined. (We are not in R2 any more, Toro!) For the operations given in each problem, decide whether
Sifting Subsets for Subspaces In each of Problems 1-3, decide whether the given subset W of the vector space V is or is not a subspace of V. If not, identify at least one requirement that is not satisfied? 1. V = R2, W = {(x, y) | y = 0} 2. V = R3, W = {(x, y) | x2 + y2 = 1} 3. V = R3, W = {(x1,
1. V = Rn, W = {x̅ ( Rn | Ax̅ = b̅, where A ( Mmn, b̅ ( 0̅} 2. V = Rn. W = {x̅ ( Rn | Ax̅ = 0̅, where A ( Mmn}?
Hyperplanes as Subspaces: The subset W of R4 defined by W = {(x, y, z, w) | ax + by + cz + dw = 0}, Where a, b, c and d are real numbers not all zero, is a hyperplane through the origin. Show that W is a subspace of R4?
Are They Subspace of Rn? Determine whether or not the subsets of Rn given in Problem 1-2 are subspaces. If not, show what required properties they fail to have? 1. Is {(a, b, a - b, a + b) | a, b ( R} a subspace of R4? 2. Is {(a, 0, b, 1, c) | a, b, c, ( R} a subspace of R5? 3. Is {(a, b, a2, b2) |
Differentiable Subspaces: Let D = D( - (, () be the vector space tractions differentiable on the real line. Which of the following subsets of I) are subspaces of D? 1. {f(t) | f' = 0} 2. [f(t) | f' = 1] 3. {f (t) | f' = f}
Property Failures Find a subset of R2 fitting each description. 1. Closed under vector addition but not under scalar multiplication? 2. Closed under scalar multiplication but not under vector addition? 3. Not closed under either vector addition or scalar multiplication?
Solution Spaces of Homogenous Linear Algebraic Systems Solve the linear systems given in Problems 1-3 and determine their solution spaces. 1. x1 - x2 + 4x4 + 2x5 - x6 = 0 2x1 - 2x2 + x3 + 2x4 + 4x5 - x6 = 0 2. 2x1 - 2x2 + 4x3 - 2x4 = 0 2x1 + x2 + 7x3 + 4x4 = 0 x1 - 4x2 - x3 + 7x4 = 0 4x1 - 12x2 -
Nonlinear Differential Equations: Show that the solution sets of the following nonlinear differential equations are not vector spaces? 1. y' = y2 2. y" + sin y = 0 3. y" + 1 / y = 0
DE Solution Spaces Recall that the "general solution" of a DE means a family or set of solution functions (y | y satisfies the DE), where each y is a function on the interval determined by the domain of the DE. In Problems 1-3, does the general solution of each of the following DEs form a vector
Line of Solution: If p̅ and h̅ are vectors in vector space V, with h̅ ( 0̅, then the line through p̅ in the direction h is defined to be the set (x̅ ( V | x̅ = p̅ + th̅ t ( R}. (a) Find the line in R2 through p̅ = (0, 1) in the direction h̅ = (2, 3). (b) Find the line in R3 through p̅
Orthogonal Complements: Prove the properties stated in Problems 1-2 using the following definition, illustrated by Fig. 3.5.4. Assume that V is a subspace of Rn.Let V be a subspace of Rn. A vector uÌ… is orthogonal to subspace V provided that uÌ… is orthogonal to every
The Spin on Spans: Determine whether the vectors in the set S span the vector space V? 1. V = R2; S = ([0, 0], [1, 1]) 2. V = R3; S = ([1, 0, 0], [0, 1, 0], [2, 3, 1]) 3. V = R3; S = ([1, 0, -1], [2, 0, 4], [-5, 0, 2], [0, 0, 1])
Function Space Dependence: Determine whether each subset S of C(R) in Problems 1-3 is or is not linearly independent? 1. S = {et, e-t} 2. S = {et, tet, t2et} 3. S = {sin t, sin 2t, sin 3t}
Independence Testing: Determine whether or not the set of functions given in each of Problems 1-3 is linearly independent on (-(, ()?1.2. 3.
Twins? Is there any difference between the vector space spanned by the set {cos t, sin t} and the vector space spanned by the set S = {cos t + sin t, cos t - sin t}?
A Questionable Basis: Is the setA basis for R3? If not, change one of the vectors to form a basis?
Wronskian Suppose that f and g are differentiable functions on the unit interval I. The Wronskian of f and g is given byShow that if W[f, g] (t) ( 0 for all t in I, then f and g are linearly independent. If c1 f(t) + c2g(t) ( 0, then c1 f' (t) + c2g'(t) ( 0; use Cramer's Rule to solve for c1 and c2?
Zero Wronskian Does Not Imply Linear Dependence: We saw that a nonzero Wronskian implies linear independence but that the converse is not true. (see Example 12.) Another Example follows:(a) Show that the Wronskian is zero for the functions(b) Show that the function f and g are not linearly
Linearly Independent Exponentials: Use the Wronskian to show that {eat, ebt} is a linearly independent set if and only if a ( b?
Looking Ahead: Use the Wronskian to show that {et, tet} is linearly independent on R?
Revisiting Linear Independence: Use the Wronskian to show that the set {et, 5e-t, e3t} is linearly independent?
Independence Checking, Use the Wronskian to check the subsets of C(R) in Problems 1-3 for linear independence? 1. {5, cos t, sin t} 2. {et, e-t, 1} 3. {2 + t, 2 - t, t2}
Getting on Base in R2: Determine whether the set given in each of Problems 1-3 is a basis for R2. Justify answers 1. {(1, 1)} 2. {[1, 2], [2, 1]} 3. {[-1, -1]. [1, 1]}
The Base for the Space: Determine whether or not the set S in each of Problem 1-3 is a basis for the specified vector space v? 1. V = R3; S = {[1, 0. 0]. [0. 1, 0]} 2. V = R3; S = {[1, 0, 1], [1, 1, 0], [0, 1, 1]} 3. V = R3; S = {[1, 0, 0] [0, 1, 0], [0, 1, 1], [1, 1, 0]}
Sizing Them Up: For each of Problems 1 and 2 determine the dimension and find a basis for the subspace W of the vector space V? 1. V = R3; W = {[x1, x2, x3] | x1 + x2 + x3 = 0} 2. V = R4; W = {[x1, x2, x3, x4] | x1, + x3 = 0, x2 = x4}
Polynomial Dimensions: Find the dimension of the subspace of P3 spanned the subset in Problems 1-2? 1. {t, t - 1} 2. {t, t - 1, t2 + 1} 3. {t2, t2 - t - i, t + 1}
Solution Basis: Determine a basis for the solution set (a subspace of R3) of the system? x + y - z = 0, y - 5z = 0
Solution Spaces for Linear Algebraic Systems In Problems 1 and 2, determine bases and dimension for the solution spaces for the homogeneous systems (as given in Section 3.5, Problems 61 and 62).1.2.
DE Solution Spaces: For each of the differential equations given in Problems 1-2, consider their solution sets. (a) Does the solution set form a subspace of the specified larger space? If not, explain? (b) If the solution set is a subspace, find a basis and determine the dimension? 1. dny/dtn = 0,
Independence Day: Decide whether the set S is a linearly independent subset of the given vector space V? 1. V = R2; S = {[1, -1], [-1, 1]} 2. V = R2; S = {[1, 1], [1, -1]} 3. V = R3; S = {[1, 0, 0], [1, 1, 0], [1, 1, 1]}
Bases for Subspaces of Rn in problems 1-3, find a basis and the dimension of the given subspaces of Rn? 1. {[a, 0, b, a - b + c] | a, b, c ( R} as a subspace of R4 2. {[a, a - b, 2a + 3b] | a, b, ( R} as a subspace of R3 3. {[x + y + z, x + y, 4z, 0] | x, y, z ( R} as a subspace of R4
Two by Two Basis Show thatIs a linearly independent set in M22. Add another vector to the set to make it a basis for M22.
Basis for Zero Trace Matrices Show thatIs a basis for the subspace of all 2 x 2 matrices in M22 with zero trace?

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