Questions and Answers of Linear Algebra

Prove Theorem 7.1(c).c. 〈u, 0〉 = 〈0, v〉 = 0
Find bases for the kernel and range of the linear transformations T in the indicated exercises. In each case, state the nullity and rank of T and verify the Rank Theorem.Exercise 4Data From Exercise
Find bases for the kernel and range of the linear transformations T in the indicated exercises. In each case, state the nullity and rank of T and verify the Rank Theorem.Exercise 3Data From Exercise
Find bases for the kernel and range of the linear transformations T in the indicated exercises. In each case, state the nullity and rank of T and verify the Rank Theorem.Exercise 2Data From Exercise
Prove Theorem 5.24.Let A be an n × n symmetric matrix. The quadratic form f (x) xTAx isa. positive definite if and only if all of the eigenvalues of A are positive.b. positive semidefinite if and
If {v1,v2,.........vn} is an orthonormal basis for Rn and A = c1v1vT1 + c2v2vT2 +..........+ cnvnvTnProve that A is a symmetric matrix with eigenvalues c1, c2,......cn and corresponding
Let (a) Orthogonally diagonalize A.(b) Give the spectral decomposition of A. 2 1 -1 A = -1 2
Find a basis for WŠ¥. -1 W = span
Find a basis for WŠ¥. W = span -1 -3 4
Find a basis for W⊥.W is the line in R3 with parametric equationsx = ty = 2tz = -t
Ifis an orthogonal matrix, find all possible values of a, b, and c. [1/2
Show thatis an orthogonal matrix. 6/7 -1/V5 _ 4/7V5 -15/7V5 2/7V5] 2/7 3/7 2/V5
Orthogonally diagonalize the matrices finding an orthogonal matrix Q and a diagonal matrix D such that QTAQ = D. A = | 0 Lo 3 1]
The coordinate vector of a vector v with respect to an orthonormal basis If find all possible vectors v. -3 [v]s 1/2. 3/5 V1 _4/5. э
Identify the quadric with the given equation and give its equation in standard form.11x2 + 11y2 + 14z2 + 2xy + 8xz - 8yz - 12x + 12y + 12z = 6
Identify the quadric with the given equation and give its equation in standard form.10x2 + 25y2 + 10z2 - 40xz + 20√2x + 50y + 20√2z = 15
Identify the quadric with the given equation and give its equation in standard form.x2 + y2 - 2z2 + 4xy - 2xz + 2yz - x + y + z = 0
Identify the quadric with the given equation and give its equation in standard form.16x2 + 100y2 + 9z2 - 24xz - 60x - 80z = 0
Identify the quadric with the given equation and give its equation in standard form.2xy + z = 0
Sometimes the graph of a quadratic equation is a straight line, a pair of straight lines, or a single point. We refer to such a graph as a degenerate conic. It is also possible that the equation is
Sometimes the graph of a quadratic equation is a straight line, a pair of straight lines, or a single point. We refer to such a graph as a degenerate conic. It is also possible that the equation is
Sometimes the graph of a quadratic equation is a straight line, a pair of straight lines, or a single point. We refer to such a graph as a degenerate conic. It is also possible that the equation is
Identify the conic with the given equation and give its equation in standard form.x2 - 2xy + y2 + 4√2x - 4 = 0
Use a rotation of axes to put the conic in standard position. Identify the graph, give its equation in the rotated coordinate system, and sketch the curve.3x2 - 2xy + 3y2 = 8
Use a translation of axes to put the conic in standard position. Identify the graph, give its equation in the translated coordinate system, and sketch the curve.2y2 - 3x2 - 18x - 20y + 11 = 0
Use a translation of axes to put the conic in standard position. Identify the graph, give its equation in the translated coordinate system, and sketch the curve.2y2 + 4x + 8y = 0
Use a translation of axes to put the conic in standard position. Identify the graph, give its equation in the translated coordinate system, and sketch the curve.x2 + 10x - 3y = -13
Identify the graph of the given equation.x = -2y2
Identify the graph of the given equation.3x2 = y2 - 1
Identify the graph of the given equation.2x2 + y2 + 8 = 0
Prove Theorem 5.25(c).c. The minimum value of f(x) is λn, and it occurs when x is a unit eigenvector corresponding to λn.
Finish the proof of Theorem 5.25(a).a. λ1 ≥ f(x) ≥ λn
Find the maximum and minimum values of the quadratic form f(x) in the given exercise, subject to the constraint ||x|| = 1, and determine the values of x for which these occur.Exercise 46Data From
Find the maximum and minimum values of the quadratic form f(x) in the given exercise, subject to the constraint ||x|| = 1, and determine the values of x for which these occur.Exercise 45Data From
Find the maximum and minimum values of the quadratic form f(x) in the given exercise, subject to the constraint ||x|| = 1, and determine the values of x for which these occur.Exercise 44Data From
Find the maximum and minimum values of the quadratic form f(x) in the given exercise, subject to the constraint ||x|| = 1, and determine the values of x for which these occur.Exercise 42Data From
Let be a symmetric 2 Ã— 2 matrix. Prove that A is positive definite if and only if a > 0 and det A > 0. -- (a-2)~ ax + 2bxy + dy = x + -y a'
Classify each of the quadratic as positive definite, positive semi definite, negative definite, negative semi definite, or indefinite.-x2 - y2 - z2 - 2xy - 2xz - 2yz
Classify each of the quadratic as positive definite, positive semi definite, negative definite, negative semi definite, or indefinite.x21 + x22 + x23 + 4x1x2
Classify each of the quadratic as positive definite, positive semi definite, negative definite, negative semi definite, or indefinite.x21 + x22 + x23 + 2x1x3
Classify each of the quadratic as positive definite, positive semi definite, negative definite, negative semi definite, or indefinite.2x21 + 2x22 + 2x23 + 2x1x2 + 2x1x3 + 2x2x3
Classify each of the quadratic as positive definite, positive semi definite, negative definite, negative semi definite, or indefinite.x2 + y2 + 4xy
Diagonalize the quadratic by finding an orthogonal matrix Q such that the change of variable x = Qy transforms the given form into one with no cross-product terms. Give Q and the new quadratic
Diagonalize the quadratic by finding an orthogonal matrix Q such that the change of variable x = Qy transforms the given form into one with no cross-product terms. Give Q and the new quadratic
Diagonalize the quadratic by finding an orthogonal matrix Q such that the change of variable x = Qy transforms the given form into one with no cross-product terms. Give Q and the new quadratic
Find the symmetric matrix A associated with the given quadratic form.2x2 - 3y2 + z2 - 4xz
Find the symmetric matrix A associated with the given quadratic form.5x21 - x22 + 2x23 + 2x1x2 - 4x1x3 + 4x2x3
Find the symmetric matrix A associated with the given quadratic form.x21 - x23 + 8x1x2 - 6x2x3
Evaluate the quadratic form f(x) = xTAx for the given A and x. 2 2 0 2 0 1,x : A = 2 %| 3 ||
Evaluate the quadratic form f(x) = xTAx for the given A and x. 1 0 -3 A = 2 -1 %3D х‑ -3 1 3
Evaluate the quadratic form f(x) = xTAx for the given A and x. 0 -3 х 1 , x =| y 2 -3 3
Find a self dual code of length 6.
Show that if C is a code with a generator matrix, then (C⊥)⊥ = C.
If C and D are codes and C ⊆ D, show that D⊥ C⊥.
Show that En and Repn are dual to each other.
(a) Find generator and parity check matrices for E3 and Rep3.(b) Show that E3 and Rep3 are dual to each other.
Find generator and parity check matrices for the dual of the (7, 4) Hamming code in Example 3.71.The even parity code En is the subset of Zn2 consisting of all vectors with even weight. The n-times
Either a generator matrix G or a parity check matrix P is given for a code C. Find a generator matrix GŠ¥and a parity check matrix PŠ¥for the dual code of C. 1 0 1 1 P = 0 1 0 0 Lo
Either a generator matrix G or a parity check matrix P is given for a code C. Find a generator matrix GŠ¥and a parity check matrix PŠ¥for the dual code of C. 1
Either a generator matrix G or a parity check matrix P is given for a code C. Find a generator matrix GŠ¥and a parity check matrix PŠ¥for the dual code of C. G =
Either a generator matrix G or a parity check matrix P is given for a code C. Find a generator matrix GŠ¥and a parity check matrix PŠ¥for the dual code of C. G =
Find the dual code CŠ¥of the code C. 1 1 C = 1 1 1 1 1
Find the dual code CŠ¥of the code C. 1 1 C = 1
Find the dual code CŠ¥of the code C. C = 1 1 [1
Find the dual code CŠ¥of the code C. _0.
P is a parity check matrix for a code C. Bring P into standard form and determine whether the corresponding code is equal to C. |P = 1 0 0 1
P is a parity check matrix for a code C. Bring P into standard form and determine whether the corresponding code is equal to C. P = | 1 1 0 0 1 0 1 1 1 Lo
P is a parity check matrix for a code C. Bring P into standard form and determine whether the corresponding code is equal to C. P =
P is a parity check matrix for a code C. Bring P into standard form and determine whether the corresponding code is equal to C.P = [1 1 0]
G is a generator matrix for a code C. Bring G into standard form and determine whether the corresponding code is equal to C. G =| 1
G is a generator matrix for a code C. Bring G into standard form and determine whether the corresponding code is equal to C. G = [1 1 1
G is a generator matrix for a code C. Bring G into standard form and determine whether the corresponding code is equal to C. 1 G =|0
G is a generator matrix for a code C. Bring G into standard form and determine whether the corresponding code is equal to C. G = | 1 11
Let q be a unit vector in Rn and let W be the subspace spanned by q. Show that the orthogonal projection of a vector v onto W (as defined in Sections 1.2 and 5.2) is given byprojW (v) = (qqT)vand
Find a symmetric 3 Ã— 3 matrix with eigenvalues Î»1, Î»2, and Î»3and corresponding orthogonal eigenvectors v1, v2, and v3. -1 A, = 1, A, = 2, 3 = 2,
Find a symmetric 2 Ã— 2 matrix with eigenvalues Î»1and Î»2and corresponding orthogonal eigenvectors v1and v2. -3 λ 2, λ = -2, Vι -2, v1 V2 3 1. ||
Find a spectral decomposition of the matrix in the given exercise.Exercise 8Data From Exercise 8 2 -1 -1 A = | -1 2 -1 1 -1 -1 2.
Find a spectral decomposition of the matrix in the given exercise.Exercise 5Data From Exercise 5 5 0 0 A = | 0 3 1
Find a spectral decomposition of the matrix in the given exercise.Exercise 2Data From Exercise 2 -2 -2
Find a spectral decomposition of the matrix in the given exercise.Exercise 1Data From Exercise 1 4 A =
If A and B are orthogonally diagonalizable and AB = BA, show that AB is orthogonally diagonalizable.
Orthogonally diagonalize the matrices finding an orthogonal matrix Q and a diagonal matrix D such that QTAQ = D. 3 2 0 3
Orthogonally diagonalize the matrices finding an orthogonal matrix Q and a diagonal matrix D such that QTAQ = D. A = 0 0 1 1 L0 0 1 1
Orthogonally diagonalize the matrices finding an orthogonal matrix Q and a diagonal matrix D such that QTAQ = D. 2 -1 -1 2 -1 -1 -1 -1 2.
Orthogonally diagonalize the matrices finding an orthogonal matrix Q and a diagonal matrix D such that QTAQ = D. -1 A = -1
Orthogonally diagonalize the matrices finding an orthogonal matrix Q and a diagonal matrix D such that QTAQ = D. [1 0 A = | 0 2] 4 2 [ 2
Orthogonally diagonalize the matrices finding an orthogonal matrix Q and a diagonal matrix D such that QTAQ = D. 6 2 A = 2 3
Orthogonally diagonalize the matrices finding an orthogonal matrix Q and a diagonal matrix D such that QTAQ = D. -2 A -2
Use the method suggested by Exercise 20 to compute A-1for the matrix A in the given exercise.Exercise 15Data From Exercise 15 1 1 0 1 1 1 1
Use the method suggested by Exercise 20 to compute A-1for the matrix A in the given exercise.Exercise 9Data From Exercise 9 1 1 0 1 1 1 1
If A is an orthogonal matrix, find a QR factorization of A.
Find a QR factorization of the matrix in the given exercise.Exercise 10Data From Exercise 10 2 -1 -1 1 5
Find a QR factorization of the matrix in the given exercise.Exercise 9Data From Exercise 9 1 1 0 1 1 [1 1
Find an orthogonal basis for 4 that contains the vectorsand 2 -1 3
Use the Gram-Schmidt Process to find an orthogonal basis for the column spaces of the matrices. 2 -1 -1 5
Find the orthogonal decomposition of v with respect to the subspace W.W as in Exercise 6Data From Exercise 6 4 X1 х — X2 Xз %| -2 Lo
Find the orthogonal decomposition of v with respect to the subspace W.W as in Exercise 5Data From Exercise 5 4 -4 3 3 X1 , X2 4 2
The given vectors form a basis for a subspace W of R3or R4. Apply the Gram-Schmidt Process to obtain an orthogonal basis for W. X1 ,X2 -2 %| Xз 1.
The given vectors form a basis for a subspace W of R3or R4. Apply the Gram-Schmidt Process to obtain an orthogonal basis for W. 3 1, X2 X1 2 4,
The given vectors form a basis for R2or R3. Apply the Gram-Schmidt Process to obtain an orthogonal basis. Then normalize this basis to obtain an orthonormal basis. X1 X2 = , Xз 1.

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