Questions and Answers of Linear Algebra

Doform a basis for Z32? 1 1 1 1 1
By considering the matrix with the given vectors as its columns.Doform a basis for R3? -1 -1 -1 1.
By considering the matrix with the given vectors as its columns.Doform a basis for R3? BIENBNE
By considering the matrix with the given vectors as its columns.Doform a basis for R3? -2 -1 -5 4 2
Give the rank and the nullity of the matrices in the given exercises.Exercise 20Data From Exercise 20 2 1 -4 0 A = -1 2 1 2 1 -2 1 4 4 3.
Give the rank and the nullity of the matrices in the given exercises.Data From Exercise 18 1 -1 A = | 1 5 -2 -1 _1
Find bases for the spans of the vectors in the given exercises from among the vectors themselves.Data from Exercise 30[3 1 -1 0], [0 -1
Find a basis for the span of the given vectors.[3 1 -1 0], [0 -1 2 -1], [4 3 8 3]
Find a basis for the span of the given vectors.[2 -3 1], [1 -1 0], [4 -4 1]
Find a basis for the span of the given vectors. 2 1 1 1 2
Explain carefully why your answers to Exercises 18 and 22 are both correct even though there appear to be differences.Data From Exercise 18 5 A = | 1 -2 -1
Find bases for row(A) and col(A) in the given exercises using AT.Data From Exercise 20 -4 0 2 A = -1 2 1 2 3 1 -2 1 4
Find bases for row(A) and col(A) in the given exercises using AT.Data From Exercise 18 -1 A = | 1 5 -1 -2
Give bases for row(A), col(A), and null(A). -4 0 2 -1 2 1 2 3 %3D 1 -2 1 4 4
Give bases for row(A), col(A), and null(A). A = | 0 1 -1 -1 -1
Give bases for row(A), col(A), and null(A). -1 A = -1 -2
If A is the matrix in Exercise 12, is in null(A)?Data From Exercise 12 -1 2 -1 |A = 0 , b = [1 -3 -3] 3 3 -1 -5
If A is the matrix in Exercise 11, is in null(A)?Data From Exercise 11 3 -1 3 = [-1 1 1]
In Exercise 12, determine whether w is in row(A) using the method described in the Remark following Example 3.41.Data From Exercise 12 -1 w = [1 -3] 3 -3 3 -5 -1 I|
Determine whether b is in col(A) and whether w is in row(A), as in Example 3.41. [1 -3 -3] 0 |, b |A = 3 %3D 3 -1 -5
Suppose S consists of all points in R2 that are on the x-axis or the y-axis (or both). (S is called the union of the two axes.) Is S a subspace of R2? Why or why not?
let S be the collection of vectorsin R3 that satisfy the given property. In each case, either prove that S forms a subspace of R3 or give a counterexample to show that it does not.|x - y| = |y
let S be the collection of vectorsin R3 that satisfy the given property. In each case, either prove that S forms a subspace of R3 or give a counterexample to show that it does not.x - y + z = 1 х
let S be the collection of vectors in R2 that satisfy the given property. In each case, either prove that S forms a subspace of R2 or give a counterexample to show that it does not.xy
Find a PTLU factorization of the given matrix A. -1 -1
Write the given permutation matrix as a product of elementary (row interchange) matrices. 1
Find an LU factorization of the given matrix. 2 2 2 1 4 -1 2 -2 6 9 5 8
Find an LU factorization of the given matrix. 1 2 3 -1 6. 3 6 -6 -1 -2 -9
Find an LU factorization of the given matrix. 2 -1 4 0 4
Solve the system Ax = b using the given LU factorization of A. 2 -4 5 -3 8 -4 4 -1 -1 -3 2 2 4 3. || 4+
Solve the system Ax = b using the given LU factorization of A. 2 -4 3 3 -1 4 2 2 A = -1 -4 2 4 , b 5 %3D -5
Solve the system Ax = b using the given LU factorization of A. -2 A = | -2 -1 3 -4 2 - 4 -3 4 -2 -3 4 -6 , b = _0 ||
Partition the given matrix so that you can apply one of the formulas and then calculate the inverse using that formula. 3 1 -1 5 2
Partition the given matrix so that you can apply one of the formulas and then calculate the inverse using that formula. -1 1 0 1 0 1. L1
Partitioning large square matrices can sometimes make their inverses easier to compute, particularly if the blocks have a nice form. Verify by block multiplication that the inverse of a matrix, if
Partitioning large square matrices can sometimes make their inverses easier to compute, particularly if the blocks have a nice form. Verify by block multiplication that the inverse of a matrix, if
Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). 5 0 2 4 over Z, 3 6 1
Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). 1 2 over Z, 2 1 _0
Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). 4 2 over Z, 4 -5 [3
Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). over Z,
Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). 1 0 0 0 La
Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). O 2V2 0 V2 V2 -4V2 3
Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). 0 -1 1 2 2 -1 3 -1
Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). Го
Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). a 0
Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). _0 1 1]
Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). 2 -1 3 2 _2 3 -1
Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). [2 0 -1 2 3
Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). 1 a - a 1
Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). 3 -4 8 9-
Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). 2 3 4 1.
Prove Theorem 3.13 for the case of AB = I.
Find the inverse of the given elementary matrix. c ,c + 0 1.
Find the inverse of the given elementary matrix. c 0 ,c + 0
Find the inverse of the given elementary matrix. 1 0 1 1 0 0
Find the inverse of the given elementary matrix. [1 0 1 -2 _0
Let In each case, find an elementary matrix E that satisfies the given equation.Is there an elementary matrix E such that EA = D? Why or why not? 2 -1 -1 B = -1 1 C = D = -3 -1 3 2 1 -1 -1 || A/
Let In each case, find an elementary matrix E that satisfies the given equation.ED = C 2 -1 -1 B = -1 1 C = D = -3 -1 3 2 1 -1 -1 || A/
Let In each case, find an elementary matrix E that satisfies the given equation.EC = D 2 -1 -1 B = -1 1 C = D = -3 -1 3 2 1 -1 -1 || A/
Let In each case, find an elementary matrix E that satisfies the given equation.EC = A 2 -1 -1 B = -1 1 C = D = -3 -1 3 2 1 -1 -1 || A/
Let In each case, find an elementary matrix E that satisfies the given equation.EA = C 2 -1 -1 B = -1 1 C = D = -3 -1 3 2 1 -1 -1 || A/
Let In each case, find an elementary matrix E that satisfies the given equation.EB = A 2 -1 -1 B = -1 1 C = D = -3 -1 3 2 1 -1 -1 || A/
Let In each case, find an elementary matrix E that satisfies the given equation.EA = B 2 -1 -1 B = -1 1 C = D = -3 -1 3 2 1 -1 -1 || A/
Solve the given matrix equation for X. Simplify your answers as much as possible. (In the words of Albert Einstein, “Everything should be made as simple as possible, but not simpler.”) Assume
Solve the given matrix equation for X. Simplify your answers as much as possible. (In the words of Albert Einstein, “Everything should be made as simple as possible, but not simpler.”) Assume
Prove Theorem 3.9(d).d. If A is an invertible matrix, then AT is invertible and(AT)-1 = (A-1)T
Prove Theorem 3.9(b). b. If A is an invertible matrix and c is a nonzero scalar, then cA is an invertible matrix and 1 (CA)¯1 = -A-1
Solve the given system using the method of Example 3.25.x1 - x2 = 2 x1 + 2x2 = 5
Find the inverse of the given matrix (if it exists) using Theorem 3.8.Where neither a, b, c, nor d is 0 1/а 1/Ь] 1/c 1/d
Find the inverse of the given matrix (if it exists) using Theorem 3.8. -b
Find the inverse of the given matrix (if it exists) using Theorem 3.8. 3.55 0.25 8.52 0.60
Find the inverse of the given matrix (if it exists) using Theorem 3.8. -4.2 -1.5 0.5 2.4
Find the inverse of the given matrix (if it exists) using Theorem 3.8. [1/V2 -1/V2] L1/V2
Find the inverse of the given matrix (if it exists) using Theorem 3.8. 5 2 3 -6 31451
Find the inverse of the given matrix (if it exists) using Theorem 3.8. 1 1
Find the inverse of the given matrix (if it exists) using Theorem 3.8. |3 4
Prove that the main diagonal of a skew-symmetric matrix must consist entirely of zeros.
Which of the following matrices are skew-symmetric?(a) (b)(c) (d) 2 -2 3 0.
Prove Theorem 3.5(b).b. For any matrix A, AAT and ATA are symmetric matrices.
Using induction, prove that for all n ≥ 1(A1 A2 ....An)T = ATn ...... AT2 AT1.
Prove Theorem 3.4(e).(Ar)T = (AT)r
Prove Theorem 3.4(a)–(c).a. (AT)T = Ab. (A + B)T = AT + BTc. (kA)T = k(AT)
Prove the half of Theorem 3.3(e) that was not proved in the text.
Prove Theorem 3.3(d).k(AB) = (kA)B = A(kB)
Prove Theorem 3.3(c).(A + B)C = AC + BC
Prove Theorem 3.2(e)–(h).e. c (A + B) = cA + cBf. (c + d)A = cA + dAg. c (dA) = (cd)Ah. 1A = A
Prove Theorem 3.2(a)–(d).a. A + B = B + Ab. (A + B) + C = A + (B + C)c. A + O = Ad. A + (-A) = O
Determine whether the given matrices are linearly independent. -1 2 4 9 2 6 0 -1 1. 0 -4
Find the general form of the span of the indicated matrices, as in Example 3.17.Span(A1, A2, A3, A4) in Exercise 8Data From Exercise 8 1 0 0 -2 5 6. 8 6 A, = B = -2 6 6. 5 A, = Аз ||
Find the general form of the span of the indicated matrices, as in Example 3.17.Span(A1, A2, A3) in Exercise 7Data From Exercise 7 -1 B = -3 A1 A2 = 2 Aз
Find the general form of the span of the indicated matrices, as in Example 3.17.Span(A1, A2, A3) in Exercise 6Data From Exercise 6 -1 B = -3 A1 A2 = 2 Aз
Write B as a linear combination of the other matrices, if possible. 6. -2 5 6. A, = B = -2 8 6. 5 6. 1 A2 A3 || || -1 A4 : -1 0.
Write B as a linear combination of the other matrices, if possible. -1 B = A, = A, = -3 2 A3
Solve the equation for X, given that A =andB =2(A - B + 2X) 3(X - B) 3 4 -1
Solve the equation for X, given that A =andB =3X = A - 2B 3 4 -1
Prove Theorem 3.1(a).
Let(a) Show that (b) Prove, by mathematical induction, that cos 0 - sin 0 sin 0 cos O cos 20 - sin 20 A? sin 20 cos 20
LetFind a formula for An (n ‰¥ 1) and verify your formula using mathematical induction. A =
LetFind, with justification, B2011. B = ||
Let(a) Compute A2, A3, . . . , A7.(b) What is A2001? Why? A = -1

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