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linear algebra

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

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. 3 -2 2 -3 1-1 4 -6 2 _2
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. 3 1
Solve the given system of equations using either Gaussian or Gauss-Jordan elimination.a + b + c + d = 10a + 2b + 3c + 4d = 30a + 3b + 6c + 10d = 65a + 4b + 8c + 15d = 93
Solve the given system of equations using either Gaussian or Gauss-Jordan elimination.w + x + 2y + z = 1w - x - y + z = 0x + y = -1w + x + z = 2
Solve the given system of equations using either Gaussian or Gauss-Jordan elimination. V2x + y + V2y – -y + V2z 2z = 3z = -V2
Solve the given system of equations using either Gaussian or Gauss-Jordan elimination. 2 6x4 + x2 - X3 - 3x4 + x = -1 X5 X1 + x2 8 - 4x5 2x3 HIN HI 61/3
Solve the given system of equations using either Gaussian or Gauss-Jordan elimination.-x1 + 3x2 - 2x3 + 4x4 = 22x1 - 6x2 + x3 - 2x4 = -1x1 - 3x2 + 4x3 - 8x4 = -4
Solve the given system of equations using either Gaussian or Gauss-Jordan elimination.2r + s = 34r + s = 72r + 5s = -1
Solve the given system of equations using either Gaussian or Gauss-Jordan elimination.3w + 3x + y = 12w + x + y + z = 12w + 3x + y - z = 2
Solve the given system of equations using either Gaussian or Gauss-Jordan elimination.x + 2y = -12x + y + z = 1-x + y - z = -1
In general, what is the elementary row operation that “undoes” each of the three elementary row operations Ri ↔ Rj, kRi, and Ri + kRj?
Use elementary row operations to reduce the given matrix to(a) Row echelon form(b) Reduced row echelon form. -2 -7 3 -9 10 -3 3
Use elementary row operations to reduce the given matrix to(a) Row echelon form(b) Reduced row echelon form. 3 -2 -1 -1 -1 [ 4 -3 -1 2.
Use elementary row operations to reduce the given matrix to(a) Row echelon form(b) Reduced row echelon form. 2 -4 -2 6 2 6. 3 -6
Use elementary row operations to reduce the given matrix to(a) Row echelon form(b) Reduced row echelon form. 3 4
Determine whether the given matrix is in row echelon form. If it is, state whether it is also in reduced row echelon form. determine whether the given matrix is in row echelon form. If it is, state whether it is also in reduced row echelon form. 2 1 3 -1 3
Determine whether the given matrix is in row echelon form. If it is, state whether it is also in reduced row echelon form. determine whether the given matrix is in row echelon form. If it is, state whether it is also in reduced row echelon form. 1 2 3
Determine whether the given matrix is in row echelon form. If it is, state whether it is also in reduced row echelon form. determine whether the given matrix is in row echelon form. If it is, state whether it is also in reduced row echelon form. 1 0 0]
Determine whether the given matrix is in row echelon form. If it is, state whether it is also in reduced row echelon form. determine whether the given matrix is in row echelon form. If it is, state whether it is also in reduced row echelon form. 1 0 3 -4 0 _0
Determine whether the given matrix is in row echelon form. If it is, state whether it is also in reduced row echelon form. determine whether the given matrix is in row echelon form. If it is, state whether it is also in reduced row echelon form. 0 0 0 0 0 0 0
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. -2a + 2(3b) = 13(2a) - 4(3b) = 1
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. x2 + 2y2 = 6x2 - y2 = 3
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. 3 = 0 || ।| ४ ला ४
Solve the linear systems in the given exercises.Data From Exercise 32 -1 0 3 1 -1 4 -1|4 2 1 1 0 2 3 0
Solve the linear systems in the given exercises.Data From Exercise 31 1 -1 2 -1
Solve the linear systems in the given exercises.Data From Exercise 30a - 2b + d = 2- a + b - c - 3d = 1
Solve the linear systems in the given exercises.Data From Exercise 29x + 5y = -1 -x + y = -5 2x + 4y = 4
Solve the linear systems in the given exercises.Data From Exercise 28 — х, %3D1 2x, + 3x, + x, = 0 Xз X1 2X3 — х, + 2x, — 2х, 3 0
Solve the linear systems in the given exercises.Data From Exercise 27x - y = 0 2x + y = 3
Find the augmented matrices of the linear systemsa - 2b + d = 2-a + b - c - 3d = 1
Solve the given system by back substitution.x - 3y + z = -5 y - 2z = -1
Solve the given system by back substitution.x1 + x2 - x3 - x4 = 1 x2 + x3 + x4 = 0 x3 - x4 = 0 x4 = 1
Solve the given system by back substitution.x1 + 2x2 + 3x3 = 0 -5x2 + 2x3 = 0 4x3 = 0
Draw graphs corresponding to the given linear systems. Determine geometrically whether each system has a unique solution, infinitely many solutions, or no solution. Then solve each system algebraically to confirm your answer.0.10x - 0.05y = 0.20-0.06x + 0.03y = -0.12
Find the solution set of each equation.4x1 + 3x2 + 2x3 = 1
Find a linear equation that has the same solution set as the given equation (possibly with some restrictions on the variables).log10 x - log10 y = 2
Determine which equations are linear equations in the variables x, y, and z. If any equation is not linear, explain why not.(cos 3) x - 4y + z = √3
Determine which equations are linear equations in the variables x, y, and z. If any equation is not linear, explain why not.3 cos x - 4y + z = √3
Determine which equations are linear equations in the variables x, y, and z. If any equation is not linear, explain why not.2x - xy - 5z = 0
Determine if the vector v is a linear combination of the remaining vectors. -1 U2 3 %| -6. ||
Mark each of the following statements true or false:(a) Every system of linear equations has a solution.(b) Every homogeneous system of linear equations has a solution.(c) If a system of linear equations has more variables than equations, then it has infinitely many solutions.(d) If a system
Consider the ISBN-10 [0, 8, 3, 7, 0, 9, 9, 0, 2, 6].(a) Show that this ISBN-10 cannot be correct.(b) Assuming that the error was a transposition error involving two adjacent entries, find the correct ISBN-10.(c) Give an example of an ISBN-10 for which a transposition error involving two
Consider the ISBN-10 [0, 4, 4, 9, 5, 0, 8, 3, 5, 6].(a) Show that this ISBN-10 cannot be correct.(b) Assuming that a single error was made and that the incorrect digit is the 5 in the fifth entry, find the correct ISBN-10.
Find the check digit d in the given International Standard Book Number (ISBN-10).[0, 3, 9, 4, 7, 5, 6, 8, 2, d]
Find the check digit d in the given International Standard Book Number (ISBN-10).[0, 3, 8, 7, 9, 7, 9, 9, 3, d]
(a) Prove that if a transposition error is made in the second and third entries of the UPC [0, 7, 4, 9, 2, 7, 0, 2, 0, 9, 4, 6], the error will be detected.(b) Show that there is a transposition involving two adjacent entries of the UPC in part (a) that would not be detected.(c) In general, when
Prove that the Universal Product Code will detect all single errors.
Consider the UPC [0, 4, 6, 9, 5, 6, 1, 8, 2, 0, 1, 5].(a) Show that this UPC cannot be correct.(b) Assuming that a single error was made and that the incorrect digit is the 6 in the third entry, find the correct UPC.
Find the check digit d in the given Universal Product Code[0, 1, 4, 0, 1, 4, 1, 8, 4, 1, 2, d]
Find the check digit d in the given Universal Product Code[0, 5, 9, 4, 6, 4, 7, 0, 0, 2, 7, d]
Prove that for any positive integers m and n, the check digit code in Znm with check vector c = 1 = [1, 1, . . . , 1] will detect all single errors. (That is, prove that if vectors u and v in Znm differ in exactly one entry, Then c · u ≠ c · v .)
Explore one approach to the problem of finding the projection of a vector onto a plane. As Figure 1.69 shows, if P is a plane through the origin in R3with normal vector n, and v is a vector in R3, then p = projP(v) is a vector in P such that v - cn = p for some scalar c.Use the method of Exercise
Explore one approach to the problem of finding the projection of a vector onto a plane. As Figure 1.69 shows, if P is a plane through the origin in R3with normal vector n, and v is a vector in R3, then p = projP(v) is a vector in P such that v - cn = p for some scalar c.Using the fact that n is
Prove equation (4) on page 44. |ax, + + byo + czo - d Va + b² + c² d(B, P) = 2
Prove equation (3) on page 43. |ax, + byo – c| Va + B d(B, €)
Find the distance between the parallel lines. х х and 1 -1
Find the point R on P that is closest to Q in Exercise 30.Data From Exercise 30Q = (0, 0, 0), P with equation x - 2y + 2z = 1
Find the point R on P that is closest to Q in Exercise 29.Data From Exercise 29.Q = (2, 2, 2), p with equation x + y - z = 0
Find the point R on „“ that is closest to Q in Exercise 28.Data From Exercise 28. -2 х Q = (0, 1, 0), l with equation | y 3
Another approach to the proof of the Cauchy-Schwarz Inequality is suggested by Figure 1.40, which shows that in R2or R3, ˆ¥proju(v) ˆ¥ ‰¤ ˆ¥vˆ¥ Show that this is equivalent to the Cauchy-Schwarz Inequality. projų(v)
Show that there are no vectors u and v such that ∥u∥ = 1, ∥v∥ = 2, and u · v = 3.
Prove Theorem 1.2(d).
Prove Theorem 1.2(b).
Solve the given equation or indicate that there is no solution.8x = 9 in Z11
Solve the given equation or indicate that there is no solution.6x = 4 in Z8
Solve the given equation or indicate that there is no solution.3x = 4 in Z6
Solve the given equation or indicate that there is no solution.3x = 4 in Z5
Find the projection of v onto u.Figure 1.39 suggests two ways in which vectors may be used to compute the area of a triangle. The area A ofthe triangle in part (a) is given by 1 / 2 ˆ¥uˆ¥ˆ¥v - proju(v)ˆ¥, and part (b) suggests the trigonometric from of the area
Find the projection of v onto u. 0.5 2.1 1.5 1.2
Find the projection of v onto u. -1 u -1 -1 -2. 2. 3. ||
Find the projection of v onto u. 2/3 -2/3 --1/3 -2 u 2. 2.
Find the projection of v onto u. Draw a sketch 3
Find the angle between u and v in the given exercise.Data From Exercise 23u = [1, 2, 3, 4], v = [5, 6, 7, 8]
Find the angle between u and v in the given exercise.Data From Exercise 22u = [1, 2, 3, 4], v = [-3, 1, 2, -2]
Find the angle between u and v in the given exercise.Data From Exercise 21u = [0.9, 2.1, 1.2], v = [-4.5, 2.6, -0.8]
Find the angle between u and v in the given exercise.Data From Exercise 20u = [5, 4, -3], v = [1, -2, -1]
Find the angle between u and v in the given exercise.Data From Exercise 18 -3
Solve the given equation or indicate that there is no solution.(a) For which values of a does x + a = 0 have a solution in Z5?(b) For which values of a and b does x + a = b have a solution in Z6?(c) For which values of a, b, and m does x + a = b have a solution in Zm?
Solve the given equation or indicate that there is no solution.6x + 3 = 1 in Z8
Solve the given equation or indicate that there is no solution.4x + 5 = 2 in Z6
Solve the given equation or indicate that there is no solution.2x + 3 = 2 in Z5
Perform the indicated calculations.[2, 0, 3, 2] · ([3, 1, 1, 2] + [3, 3, 2, 1]) in Z44 and in Z45
Perform the indicated calculations.[2, 1, 2] · [2, 2, 1] in Z33
Perform the indicated calculations.[2, 1, 2] + [2, 0, 1] in Z33
Perform the indicated calculations.2100 in Z11
Perform the indicated calculations.8(6 + 4 + 3) in Z9
Perform the indicated calculations.(3 + 4)(3 + 2 + 4 + 2) in Z5
Perform the indicated calculations.2 + 1 + 2 + 2 + 1 in Z3, Z4, and Z5
Perform the indicated calculations.3(3 + 3 + 2) in Z4
If possible, solve 3(x + 2) = 5 in Z9.
Refer to check digit codes in which the check vector is the vector c = [1, 1, . . . , 1] of the appropriate length. In each case, find the check digit d that would be appended to the vector u.u = [3, 0, 7, 5, 6, 8] in Z69
Refer to check digit codes in which the check vector is the vector c = [1, 1, . . . , 1] of the appropriate length. In each case, find the check digit d that would be appended to the vector u.u [1, 5, 6, 4, 5] in Z57
Refer to check digit codes in which the check vector is the vector c = [1, 1, . . . , 1] of the appropriate length. In each case, find the check digit d that would be appended to the vector u.u = [3, 4, 2, 3] in Z45
Refer to check digit codes in which the check vector is the vector c = [1, 1, . . . , 1] of the appropriate length. In each case, find the check digit d that would be appended to the vector u.u = [1, 2, 2, 2] in Z43
A parity check code vector v is given. Determine whether a single error could have occurred in the transmission of v.v = [1, 1, 0, 1, 0, 1, 1, 1]
A parity check code vector v is given. Determine whether a single error could have occurred in the transmission of v.v = [0, 1, 0, 1, 1, 1]
A parity check code vector v is given. Determine whether a single error could have occurred in the transmission of v.v = [1, 1, 1, 0, 1, 1]
A parity check code vector v is given. Determine whether a single error could have occurred in the transmission of v.v = [1, 0, 1, 0]
Find the parity check code vector for the binary vector u.u = [1, 1, 0, 1, 1]
Find the parity check code vector for the binary vector u.u = [1, 0, 1, 1]

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