Questions and Answers of Linear Algebra

Use the method of Example 6.83 to evaluate the given integral.∫ (x cos x + x sin x) dx. (See Exercise 16.)Data From Exercise 16Consider the subspace W of D, given by W = span(cos x, sin x, x cos x,
Which of the codes are linear codes?The odd parity code On consisting of all vectors in Zn2 with odd weight.
Use Theorem 6.2 to determine whether W is a subspace of V.V = Mnn, W = {A in Mnn : det A = 1}
(a) Show that e[0, 1] ≅ e[2, 3]. [Define T : e[0, 1] → e[2, 3] by letting T(f ) be the function whose value at x is (T(f)) (x) = f(x -2) for x in [2, 3].](b) Show that e[0, 1] ≅ e[a, a + 1] for
Find the other two bases for the code C1 in Example 6.94.
Let B be a basis for a vector space V, let u1, . . . , uk be vectors in V, and let c1, . . . , ck be scalars. Show that [c1u1 +.......+ ckuk]B = c1[u1]B +..... + ck[uk]B.
Use Theorem 6.2 to determine whether W is a subspace of V.V = Mnn, W is the set of diagonal n × n matrices
Show that e|0, 1| ≅ e [0,2].
Use Theorem 6.2 to determine whether W is a subspace of V.V = Mnn, W is the set of idempotent n × n matrices
Test the sets of polynomials for linear independence. For those that are linearly dependent, express one of the polynomials as a linear combination of the others.{x, 2x - x2, 3x + 2x2} in P2
Use Theorem 6.2 to determine whether W is a subspace of V.V = Mnn, W = {A in Mnn : AB = BA}, where B is a given (fixed) matrix
Use Theorem 6.2 to determine whether W is a subspace of V.V = P2, W = {bx + cx2}
Use Theorem 6.2 to determine whether W is a subspace of V.V = P2, W = {a + bx + cx2 : a + b + c = 0}
Use Theorem 6.2 to determine whether W is a subspace of V.V = P, W is the set of all polynomials of degree 3.
Find a parity check matrix for R2.
Use Theorem 6.2 to determine whether W is a subspace of V.V = F, W = {f in F : f (-x) = f(x)}
Find a parity check matrix for R3.
Use Theorem 6.2 to determine whether W is a subspace of V.V = F, W is the set of all integrable functions
Use Theorem 6.2 to determine whether W is a subspace of V.V = D, W = {f in D : f′ (x) ≥ 0 for all x}
Let T : V → W be a linear transformation between finite-dimensional vector spaces V and W. Let B and C be bases for V and W, respectively, and let A = [T]C←B.Use the results of this section to
Use Theorem 6.2 to determine whether W is a subspace of V.V = F, W = e(2), the set of all functions with continuous second derivatives.
Extendto a basis for M22. 1 1 0 1
Suppose that u, v, and w are vectors in an inner product space such that{u, v} = 1, {u, w} = 5, {v, w} = 0||u|| = 1, ||v|| = √3, ||w|| = 2Evaluate the expressionsShow that u + v = w [How can you
Prove Theorem 7.15(d).
Draw a picture (similar to Figure 7.28) to illustrate Example 7.45. y C1 хе Co (b) (a) Figure 7.28
Find the standard matrix of the orthogonal projection onto the subspace W. Then use this matrix to find the orthogonal projection of v onto W. W = span |, v =
Compute the pseudo inverse A+of A in the given exercise.Exercise 8Data From Exercise 8 A = -2 2
Let C be a linear code. Show that the minimum distance of C is equal to the minimum weight of a nonzero code vector.
Compute the pseudo inverse A+of A in the given exercise.Exercise 9Data From Exercise 9 2 0 A =
Show that d - 1 ≤ n - k for any linear (n, k, d) code.
Let and (a) Compute condˆž (A).(b) Suppose A is changed toHow large a relative change can this change produce in the solution to Ax = b?(c) Solve the systems using A and A' and
Verify that the standard matrix of the projection onto W in Example 7.31 (as constructed by Theorem 7.11) does not depend on the choice of basis. Takeas a basis for W and repeat the calculations to
Compute the pseudo inverse A+of A in the given exercise.Exercise 10Data From Exercise 10 A = | 0 -3 0
Let C be a linear (n, k, d) code with parity check matrix P. Prove that d = n - k + 1 if and only if every n - k columns of P are linearly independent.
Let and(a) Compute cond1 (A).(b) Suppose A is changed to How large a relative change can this change produce in the solution to Ax = b?(c) Solve the systems using A and A' and determine the
Show that (A+)+ = A. [Show that A satisfies the Penrose conditions for A+. By Exercise 52, A must therefore be (A+)+.]
Compute the pseudo inverse of A. A =
Prove the remaining part of Theorem 7.12(c).
Prove that if A is a positive definite matrix with SVD A = U ∑VT, then U = V.
Find a polar decomposition of the matrices in A in Exercise 3Data From Exercise 3 A =
Find a polar decomposition of the matrices in A in Exercise 14Data From Exercise 14 A = - 3 -1
Find a polar decomposition of the matrices in A in Exercise 14Data From Exercise 14 -3 4 2 A = -2 2 6. %D 4 -1 6.
Compute the pseudo inverse of A. 2 -1 1 -2 2
Write the given system in the form of equation (7). Then use the method of Example 7.22 to estimate the number of iterations of Jacobi’s method that will be needed to approximate the solution to
Compute the pseudo inverse of A. 1 0 1 A =
Write the given system in the form of equation (7). Then use the method of Example 7.22 to estimate the number of iterations of Jacobi’s method that will be needed to approximate the solution to
Compute the pseudo inverse of A. 2 3 4
Write the given system in the form of equation (7). Then use the method of Example 7.22 to estimate the number of iterations of Jacobi’s method that will be needed to approximate the solution to
Compute the pseudo inverse of A. A =
(a) Set up and solve the normal equations for the system of equations in Example 7.40.(b) Find a parametric expression for the length of a solution vector in part (a).(c) Find the solution vector of
Compute the pseudo inverse of A. 3 A = | 3 1 2 2
Write the given system in the form of equation (7). Then use the method of Example 7.22 to estimate the number of iterations of Jacobi’s method that will be needed to approximate the solution to
Compute the pseudo inverse of A. 3 A = | -1
Compute the pseudo inverse of A. A = -1
Show that if A and B are invertible matrices, then cond (AB) ≤ cond (A) cond (B) with respect to any matrix norm.
(a) If a (9, 4) linear code has generator matrix G and parity check matrix P, what are the dimensions of G and P?(b) Repeat part (a) for an (n, k) linear code.
Show that e[a, b] ≅ e [c,d] for all a < b and c < d.
A linear transformation T : V → V is given. If possible, find a basis C for V such that the matrix [T]c of T with respect to C is diagonal.T : P1 → P1 defined by T(a + bx) = (4a + 2b) + (a +
For a linear code C, show that (C⊥)⊥ without using matrices.
A linear transformation T : V → V is given. If possible, find a basis C for V such that the matrix [T]c of T with respect to C is diagonal.T : P2 → P2 defined by T(p(x)) = p(x + 1)
If C is an (n, k) linear code that is self dual, prove that n must be even. [Use the analogue in Zn2 of Theorem 5.13.]
A linear transformation T : V → V is given. If possible, find a basis C for V such that the matrix [T]c of T with respect to C is diagonal.T : P1 → P1 defined by T(p(x)) = p(x) + xp′(x)
Write out the vectors in the Reed-Muller code R3.
Use Theorem 6.2 to determine whether W is a subspace of V.V = P2, W = {a + bx + cx2 : abc = 0}
A linear transformation T : V → V is given. If possible, find a basis C for V such that the matrix [T]c of T with respect to C is diagonal.T : P2 → P2 defined by T(p(x)) = p(3x + 2)
Define a family of matrices inductively as follows:G0= [1] and, for n ‰¥ 1,where 0 is a zero vector and 1 is a vector consisting entirely of ones.(a) Write out G1, G2,
Find the dimension of the vector space V and give a basis for V.V = {A in M22 : A is skew-symmetric}
Use Theorem 6.2 to determine whether W is a subspace of V.V = F, W = {f in F : f (-x) = -f (x)}
Prove that, for a linear code C, either all the code vectors have even weight or exactly half of them do. [Let E be the set of vectors in C with even weight and O the set of vectors in C with odd
Find the dimension of the vector space V and give a basis for V.V = {A in M22 : AB = BA}, Where B =
Use Theorem 6.2 to determine whether W is a subspace of V.V = F, W = {f in F : f (0) = 1}
Use Theorem 6.2 to determine whether W is a subspace of V.V = F, W = {f in F : f (0) = 0}
Let T : V → W be a linear transformation between finite-dimensional vector spaces V and W. Let B and C be bases for V and W, respectively, and let A = [T]C←B.If B′ and C′ are also bases for V
Use Theorem 6.2 to determine whether W is a subspace of V.V = F, W = {f in F : lim f(x) = 0} %3|
Let T : V → W be a linear transformation between finite-dimensional vector spaces V and W. Let B and C be bases for V and W, respectively, and let A = [T]C←B.If dim V = n and dim W = m, prove
If V is a vector space, then the dual space of V is the vector space V* = L (V, R). Prove that if V is finite dimensional, then V* ≅ V.
Extendto a basis for M22. 1 1 1 1
Extendto a basis for the vector space of symmetric 2 Ã— 2 matrices. 0 1 1 0
Find a basis for span 1 1 1 1 1
let f(x) = sin2x and g(x) = cos2x. Determine whether h(x) is in span(f(x), g(x)).h(x) = 1
Find the singular values of the given matrix. 3 A = 3
Consider the data points (1, 0), (2, 1), and (3, 5). Compute the least squares error for the given line. In each case, plot the points and the line.y = -3 + 2x
Let (u, v) is the inner product of Example 7.3 withCompute(a) (u, v) (b) ||u|| (c) d(u, v) 2 3 and v = 4 %3D -1 |A = 7.
Find the singular values of the given matrix. [V2 A = V2.
Find the best linear approximation to f on the interval [-1, l].F(x) = sin(πx/2)
Consider the data points (-5, 3), (0, 3), (5, 2), and (10, 0). Compute the least squares error for the given line. In each case, plot the points and the line.y = 2 - x
Determine whether the definition gives an inner product. i 8) = („max f(x))(,max g(x)) for f, g in C[0, 1] 0
In Exercise 2, find a nonzero vector orthogonal to u.Data From Exercise 2(u, v) is the inner product of Example 7.3 withCompute(a) (u, v) (b) ||u|| (c) d(u, v) |A = 7.
Find the singular values of the given matrix. 3 A = 4.
Let p(x) = 2 - 3x + x2 and q(x) = 1 - 3x2. compute (a) {p(x), q(x)} (b) ||p(x)||(c) d(p(x), q(x)){p(x), q(x)} is the inner product of Example 7.4.
Find the singular values of the given matrix.A = [3 4]
Consider the data points (-5, 3), (0, 3), (5, 2), and (10, 0). Compute the least squares error for the given line. In each case, plot the points and the line.y = 2 - 1/5x
Let p(x) = 2 - 3x + x2 and q(x) = 1 - 3x2. compute (a) {p(x), q(x)} (b) ||p(x)||(c) d(p(x), q(x)){p(x), q(x)} is the inner product of Example 7.5 on the vector space P2 [0, 1].
Find the singular values of the given matrix. -2 0
In Exercise 5, find a nonzero vector orthogonal to p(x).Data From Exercise 5{p(x), q(x)} is the inner product of Example 7.4.
Find the singular values of the given matrix. A = -2 2
Find the least squares approximating line for the given points and compute the corresponding least squares error.(1, 5), (2, 3), (3, 2)
In Exercise 6, find a nonzero vector orthogonal to p(x).Data From Exercise 6{p(x), q(x)} is the inner product of Example 7.5 on the vector space P2 [0, 1].
Find the singular values of the given matrix. 0 1 2 A = [0 2 0]
Find the least squares approximating line for the given points and compute the corresponding least squares error.(0, 4), (1, 1), (2, 0)

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