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linear algebra

Linear Algebra A Modern Introduction 3rd edition David Poole - Solutions

Letand Compute the outer product expansion of BA. -2 A = -3 2 0 -1 3 0 B = -1 6. 4.
Letand Use the row-matrix representation of the product to write each row of BA as a linear combination of the rows of A. -2 A = -3 2 0 -1 3 0 B = -1 6. 4.
Letand Use the matrix-column representation of the product to write each column of BA as a linear combination of the columns of B. -2 A = -3 2 0 -1 3 0 B = -1 6. 4.
Letand Compute the outer product expansion of AB. -2 A = -3 2 0 -1 3 0 B = -1 6. 4.
Letand Use the row-matrix representation of the product to write each row of AB as a linear combination of the rows of B. -2 A = -3 2 0 -1 3 0 B = -1 6. 4.
Letand Use the matrix-column representation of the product to write each column of AB as a linear combination of the columns of A. -2 A = -3 2 0 -1 3 0 B = -1 6. 4.
LetCompute the indicated matrices (if possible).(I2 - A)2 2 1 C = 3 |A = 4 B = -2 3 4 2 3 5 5 6 -3 E = [4 2], F= D = %3D 2.
LetCompute the indicated matrices (if possible).A3 2 1 C = 3 |A = 4 B = -2 3 4 2 3 5 5 6 -3 E = [4 2], F= D = %3D 2.
LetCompute the indicated matrices (if possible).DA - AD 2 1 C = 3 |A = 4 B = -2 3 4 2 3 5 5 6 -3 E = [4 2], F= D = %3D 2.
LetCompute the indicated matrices (if possible).BTCT - (CB)T 2 1 C = 3 |A = 4 B = -2 3 4 2 3 5 5 6 -3 E = [4 2], F= D = %3D 2.
LetCompute the indicated matrices (if possible).EF 2 1 C = 3 |A = 4 B = -2 3 4 2 3 5 5 6 -3 E = [4 2], F= D = %3D 2.
LetCompute the indicated matrices (if possible).FE 2 1 C = 3 |A = 4 B = -2 3 4 2 3 5 5 6 -3 E = [4 2], F= D = %3D 2.
LetCompute the indicated matrices (if possible).F(AF) 2 1 C = 3 |A = 4 B = -2 3 4 2 3 5 5 6 -3 E = [4 2], F= D = %3D 2.
LetCompute the indicated matrices (if possible).E(AF) 2 1 C = 3 |A = 4 B = -2 3 4 2 3 5 5 6 -3 E = [4 2], F= D = %3D 2.
LetCompute the indicated matrices (if possible).BT B 2 1 C = 3 |A = 4 B = -2 3 4 2 3 5 5 6 -3 E = [4 2], F= D = %3D 2.
LetCompute the indicated matrices (if possible).D + BC 2 1 C = 3 |A = 4 B = -2 3 4 2 3 5 5 6 -3 E = [4 2], F= D = %3D 2.
LetCompute the indicated matrices (if possible).B2 2 1 C = 3 |A = 4 B = -2 3 4 2 3 5 5 6 -3 E = [4 2], F= D = %3D 2.
Compute the indicated matrices (if possible).AB 2 1 C = 3 |A = 4 B = -2 3 4 2 3 5 5 6 -3 E = [4 2], F= D = %3D 2.
LetCompute the indicated matrices (if possible).B - CT 2 1 C = 3 |A = 4 B = -2 3 4 2 3 5 5 6 -3 E = [4 2], F= D = %3D 2.
LetCompute the indicated matrices (if possible).2D - 5A 2 1 C = 3 |A = 4 B = -2 3 4 2 3 5 5 6 -3 E = [4 2], F= D = %3D 2.
Let a1, a2, a3 be linearly independent vectors in R3, and let A = [a1 a2 a3]. Which of the following statements are true?(a) The reduced row echelon form of A is I3.(b) The rank of A is 3.(c) The system [A | b] has a unique solution for any vector b in
Determine whether R3= span(u, v, w) if:(a)(b) u = , v = | 0 -1 -1|,v= -1
Determine whetherare linearly independent. 2 3 -1 -2, -3 -2.
Determine whetheris in the span ofand 3 5 -1 3
Find the point of intersection of the following lines, if it exists.and х 2 + s -1 2 || х 5 -2| + t -4.
Draw diagrams to illustrate the convergence of the Gauss-Seidel method with the given system.2x1 + x2 = 5x1 + x2 = 1
Draw diagrams to illustrate the convergence of the Gauss-Seidel method with the given system.7x1 - x2 = 6x1 - 5x2 = -4
Repeat the given exercise using the GaussSeidel method. Take the zero vector as the initial approximation and work with four-significant-digit accuracy until two successive iterates agree within 0.001 in each variable. Compare the number of iterations required by the Jacobi and Gauss-Seidel methods
Repeat the given exercise using the GaussSeidel method. Take the zero vector as the initial approximation and work with four-significant-digit accuracy until two successive iterates agree within 0.001 in each variable. Compare the number of iterations required by the Jacobi and Gauss-Seidel methods
Repeat the given exercise using the GaussSeidel method. Take the zero vector as the initial approximation and work with four-significant-digit accuracy until two successive iterates agree within 0.001 in each variable. Compare the number of iterations required by the Jacobi and Gauss-Seidel methods
Apply Jacobi’s method to the given system. Take the zero vector as the initial approximation and work with four-significant-digit accuracy until two successive iterates agree within 0.001 in each variable. In each case, compare your answer with the exact solution found using any direct method you
Apply Jacobi’s method to the given system. Take the zero vector as the initial approximation and work with four-significant-digit accuracy until two successive iterates agree within 0.001 in each variable. In each case, compare your answer with the exact solution found using any direct method you
Apply Jacobi’s method to the given system. Take the zero vector as the initial approximation and work with four-significant-digit accuracy until two successive iterates agree within 0.001 in each variable. In each case, compare your answer with the exact solution found using any direct method you
The process of adding rational functions (ratios of polynomials) by placing them over a common denominator is the analogue of adding rational numbers. The reverse process of taking a rational function apart by writing it as a sum of simpler rational functions is useful in several areas of
The process of adding rational functions (ratios of polynomials) by placing them over a common denominator is the analogue of adding rational numbers. The reverse process of taking a rational function apart by writing it as a sum of simpler rational functions is useful in several areas of
The process of adding rational functions (ratios of polynomials) by placing them over a common denominator is the analogue of adding rational numbers. The reverse process of taking a rational function apart by writing it as a sum of simpler rational functions is useful in several areas of
The process of adding rational functions (ratios of polynomials) by placing them over a common denominator is the analogue of adding rational numbers. The reverse process of taking a rational function apart by writing it as a sum of simpler rational functions is useful in several areas of
The process of adding rational functions (ratios of polynomials) by placing them over a common denominator is the analogue of adding rational numbers. The reverse process of taking a rational function apart by writing it as a sum of simpler rational functions is useful in several areas of
The process of adding rational functions (ratios of polynomials) by placing them over a common denominator is the analogue of adding rational numbers. The reverse process of taking a rational function apart by writing it as a sum of simpler rational functions is useful in several areas of
The process of adding rational functions (ratios of polynomials) by placing them over a common denominator is the analogue of adding rational numbers. The reverse process of taking a rational function apart by writing it as a sum of simpler rational functions is useful in several areas of
Set up and solve an appropriate system of linear equations to answer the questions.Through any three noncollinear points there also passes a unique circle. Find the circles (whose general equations are of the form x2 + y2 + ax + by + c = 0) that pass through the sets of points in Exercise 45. (To
Set up and solve an appropriate system of linear equations to answer the questions.From elementary geometry we know that there is a unique straight line through any two points in a plane. Less well known is the fact that there is a unique parabola through any three noncollinear points in a plane.
Set up and solve an appropriate system of linear equations to answer the questions.Generalizing Exercise 42, find conditions on the entries of a 3×3 addition table that will guarantee that we can solve for a, b, c, d, e, and f as previously.
Set up and solve an appropriate system of linear equations to answer the questions.Describe all possible values of a, b, c, d, e, and f that will make each of the following a valid addition table.(a)(b) b. a 3 2 1 5 4 3 4 3 2 3 3 4 5 4 5 6
Set up and solve an appropriate system of linear equations to answer the questions.What conditions on w, x, y, and z will guarantee that we can find a, b, c, and d so that the following is a valid addition table? х
Set up and solve an appropriate system of linear equations to answer the questions.Describe all possible values of a, b, c, and d that will make each of the following a valid addition table.a)b) 2 3 d | 4 5 a 3 6 4 5
Set up and solve an appropriate system of linear equations to answer the questions.There are three types of corn. Three bundles of the first type, two of the second, and one of the third make 39 measures. Two bundles of the first type, three of the second, and one of the third make 34 measures. And
(a) Set up and solve a system of linear equations to find the possible flows in the network shown in Figure 2.21.(b) Is it possible for f1= 100 and f6= 150? (Answer this question first with reference to your solution in part (a) and then directly from Figure 2.21.)(c) If f4= 0, what will the range
A network of irrigation ditches is shown in Figure 2.20, with flows measured in thousands of liters per day(a) Set up and solve a system of linear equations to find the possible flows f1, . . . , f5.(b) Suppose DC is closed. What range of flow will need to be maintained through DB?(c) From
The downtown core of Gotham City consists of one-way streets, and the traffic flow has been measured at each intersection. For the city block shown in Figure 2.19, the numbers represent the average numbers of vehicles per minute entering and leaving intersections A, B, C, and D during business
Balance the chemical equation for each reaction.C2H2Cl4 + Ca(OH)2 → C2HCl3 + CaCl2 + H2O
Balance the chemical equation for each reaction.Na2CO3 + C + N2 → NaCN + CO
Balance the chemical equation for each reaction.HClO4 + P4O10 → H3PO4 + Cl2O7
Balance the chemical equation for each reaction.C5H11OH + O2 → H2O + CO2 (This equation represents the combustion of amyl alcohol.)
Balance the chemical equation for each reaction.C7H6O2 + O2 → H2O + CO2
Balance the chemical equation for each reaction.C4H10 + O2 → CO2 + H2O (This reaction occurs when butane, C4H10, burns in the presence of oxygen to form carbon dioxide and water.)
Balance the chemical equation for each reaction.CO2 + H2O → C6H12O6 + O2 (This reaction takes place when a green plant converts carbon dioxide and water to glucose and oxygen during photosynthesis.)
Balance the chemical equation for each reaction.FeS2 + O2 → Fe2O3 + SO2
The systems of equations are nonlinear. Find substitutions (changes of variables) that convert each system into a linear system and use this linear system to help solve the given system. tan x - 2 sin y = 2tan x - sin y + cos z = 2
Determine if the sets of vectors in the given exercise are linearly independent by converting the vectors to row vectors and using the method of Example 2.25 and Theorem 2.7. For any sets that are linearly dependent, find a dependence relationship among the vectors.Exercise 31Data From Exercise 31
Determine if the sets of vectors in the given exercise are linearly independent by converting the vectors to row vectors and using the method of Example 2.25 and Theorem 2.7. For any sets that are linearly dependent, find a dependence relationship among the vectors.Exercise 30Data From Exercise 30
Determine if the sets of vectors in the given exercise are linearly independent by converting the vectors to row vectors and using the method of Example 2.25 and Theorem 2.7. For any sets that are linearly dependent, find a dependence relationship among the vectors.Exercise 29Data From Exercise 29
Determine if the sets of vectors in the given exercise are linearly independent by converting the vectors to row vectors and using the method of Example 2.25 and Theorem 2.7. For any sets that are linearly dependent, find a dependence relationship among the vectors.Exercise 28Data From Exercise 28
Determine if the sets of vectors in the given exercise are linearly independent by converting the vectors to row vectors and using the method of Example 2.25 and Theorem 2.7. For any sets that are linearly dependent, find a dependence relationship among the vectors.Exercise 27Data From Exercise 27
Determine if the sets of vectors in the given exercise are linearly independent by converting the vectors to row vectors and using the method of Example 2.25 and Theorem 2.7. For any sets that are linearly dependent, find a dependence relationship among the vectors.Exercise 26Data From Exercise 26
Determine if the sets of vectors in the given exercise are linearly independent by converting the vectors to row vectors and using the method of Example 2.25 and Theorem 2.7. For any sets that are linearly dependent, find a dependence relationship among the vectors.Exercise 25Data From Exercise 25
Determine if the sets of vectors in the given exercise are linearly independent by converting the vectors to row vectors and using the method of Example 2.25 and Theorem 2.7. For any sets that are linearly dependent, find a dependence relationship among the vectors.Exercise 24Data From Exercise 24
Determine if the sets of vectors in the given exercise are linearly independent by converting the vectors to row vectors and using the method of Example 2.25 and Theorem 2.7. For any sets that are linearly dependent, find a dependence relationship among the vectors.Exercise 23Data From Exercise 23
Determine if the sets of vectors in the given exercise are linearly independent by converting the vectors to row vectors and using the method of Example 2.25 and Theorem 2.7. For any sets that are linearly dependent, find a dependence relationship among the vectors.Exercise 22Data From Exercise 22
Use the method of Example 2.23 and Theorem 2.6 to determine if the sets of vectors in Exercises 22€“31 are linearly independent. If, or any of these, the answer can be determined by inspection (i.e., without calculation), state why. For any sets that are linearly dependent, find a
Use the method of Example 2.23 and Theorem 2.6 to determine if the sets of vectors in Exercises 22€“31 are linearly independent. If, or any of these, the answer can be determined by inspection (i.e., without calculation), state why. For any sets that are linearly dependent, find a
Use the method of Example 2.23 and Theorem 2.6 to determine if the sets of vectors in Exercises 22€“31 are linearly independent. If, or any of these, the answer can be determined by inspection (i.e., without calculation), state why. For any sets that are linearly dependent, find a
Use the method of Example 2.23 and Theorem 2.6 to determine if the sets of vectors in Exercises 22€“31 are linearly independent. If, or any of these, the answer can be determined by inspection (i.e., without calculation), state why. For any sets that are linearly dependent, find a
Use the method of Example 2.23 and Theorem 2.6 to determine if the sets of vectors in Exercises 22€“31 are linearly independent. If, or any of these, the answer can be determined by inspection (i.e., without calculation), state why. For any sets that are linearly dependent, find a
Use the method of Example 2.23 and Theorem 2.6 to determine if the sets of vectors in Exercises 22€“31 are linearly independent. If, or any of these, the answer can be determined by inspection (i.e., without calculation), state why. For any sets that are linearly dependent, find a
Use the method of Example 2.23 and Theorem 2.6 to determine if the sets of vectors in Exercises 22€“31 are linearly independent. If, or any of these, the answer can be determined by inspection (i.e., without calculation), state why. For any sets that are linearly dependent, find a
Use the method of Example 2.23 and Theorem 2.6 to determine if the sets of vectors in Exercises 22€“31 are linearly independent. If, or any of these, the answer can be determined by inspection (i.e., without calculation), state why. For any sets that are linearly dependent, find a
Use the method of Example 2.23 and Theorem 2.6 to determine if the sets of vectors in Exercises 22€“31 are linearly independent. If, or any of these, the answer can be determined by inspection (i.e., without calculation), state why. For any sets that are linearly dependent, find a
Describe the span of the given vectors(a) Geometrically(b) Algebraically -1 -1 1. -1
Show that R3= span 1 1 1 1 1
Show that R3= span L0]
Show that R2 = Span 2 2
Show that R2= span -1
Determine if the vector b is in the span of the columns of the matrix A. 1 2 3 10 b =| 11 4 5 6 7 8 9 12
Determine if the vector v is a linear combination of the remaining vectors. 2.4 2.2 1.0 0.5 , u2 4.0 , uj 1.2 [ 3.1 -2.2 -0.5 1.2 - 2.3 uz 4.8 ||
Determine if the vector v is a linear combination of the remaining vectors. 3 uj u2 1 -2
When p is not prime, extra care is needed in solving a linear system (or, indeed, any equation) over Zp. Using Gaussian elimination, solve the following system over Z6. What complications arise?2x+ 3y = 44x + 3y = 2
Prove the following corollary to the Rank Theorem:Let A be an m × n matrix with entries in Zp. Any consistent system of linear equations with coefficient matrix A has exactly pn-rank(A) solutions over Zp.
Solve the systems of linear equations over the indicated Zp.x1 + 4x4 = 1 over Z5x1 + 2x2 + 4x3 = 32x1 + 2x2 + x4 = 1x1 + 3x3 = 2
Solve the systems of linear equations over the indicated Zp.3x + 2y = 1 over Z7x + 4y = 1
Solve the systems of linear equations over the indicated Zp.3x + 2y = 1 over Z5x + 4y = 1
Solve the systems of linear equations over the indicated Zp.x + 2y = 1 over Z3x + y = 2
Find the line of intersection of the given planes.4x + y - z = 0 and 2x - y + 3z = 4
Give examples of homogeneous systems of m linear equations in n variables with m = n and with m > n that have (a) Infinitely many solutions and (b) A unique solution.
What value(s) of k, if any, will the systems have(a) No solution,(b) A unique solution,(c) Infinitely many solutions?x + y + kz =1x + ky + z = 1kx + y + z = -2
What value(s) of k, if any, will the systems have(a) No solution,(b) A unique solution,(c) Infinitely many solutions?x + y + z = 2 x + 4y - z = k 2x - y + 4z = k2
What value(s) of k, if any, will the systems have (a) no solution, (b) a unique solution, and (c) infinitely many solutions?kx + y = -22x - 2y = 4
Show that if ad - bc ≠ 0, then the systemax + by = r cx + dy = shas a unique solution.
Determine by inspection (i.e., without performing any calculations) whether a linear system with he given augmented matrix has a unique solution, infinitely many solutions, or no solution. Justify your answers. 5 6 4 2 3 3 21 5 4 6. 7 7 7. _7
Determine by inspection (i.e., without performing any calculations) whether a linear system with he given augmented matrix has a unique solution, infinitely many solutions, or no solution. Justify your answers. 4 0 3 5 6. 12 0 _9 10 11

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