1 Million+ Step-by-step solutions

Write the equation of the line passing through P with normal vector n in (a) normal form and (b) general form.

Find the distance between the parallel lines.

a) Prove that if a transposition error is made in the fourth and fifth entries of the ISBN-10 [0, 6, 7, 9, 7, 6, 2, 9, 0, 6], the error will be detected.

b) Prove that if a transposition error is made in any two adjacent entries of the ISBN-10 in part (a), the error will be detected.

c) Prove, in general, that the ISBN-10 code will always detect a transposition error involving two adjacent entries.

Is M_{22}spanned by

Is M_{22 }spanned by

Determine whether V and W are isomorphic. If they are, give an explicit isomorphism T : V → W.

V = P_{2}, W = {p{x} in P_{3} : p(0) = 0}

Determine whether the set B is a basis for the vector space V.

V = P_{2}, B = {2 + 3x - x^{2}, 1 + 2x + 2x^{2}}

Which of the codes are linear codes?

Prove Theorem 6.1(c).

Let V be a vector space, u a vector in V, and c a scalar.

a. 0u 0

b. c0 0

c. (-1)u-u

d. If cu 0, then c 0 or u 0.

Determine whether the set B is a basis for the vector space V.

V = P_{2}, B = {1 - x, 1 - x^{2}, x - x^{2}}

Which of the codes are linear codes?

Prove Theorem 6.1(a).

Let V be a vector space, u a vector in V, and c a scalar.

a. 0u = 0

b. c0 = 0

c. (-1)u = -u

d. If cu = 0, then c = 0 or u = 0.

Determine whether the set B is a basis for the vector space V.

V = P_{2}, B = {x, 1 - x, 1 + x + x^{2}}

Prove Theorem 6.14(b).

Let T : V→W be a linear transformation. Then:

T(-v) = - T(v) for all v in V.

Determine whether the given set, together with the specified operations of addition and scalar multiplication, is a vector space over the indicated Z_{p}. If it is not, list all of the axioms that fail to hold.

Z_{6}, over Z_{3} with the usual addition and multiplication (Think this one through carefully!)

Determine whether the set B is a basis for the vector space V.

Prove Theorem 6.33(b).

Let S be the solution space of

y'' + ay' + by = 0

and let λ_{1} and λ_{2} be the roots of the characteristic equation λ^{2} + aλ + b = 0.

b. If λ_{1} = λ_{2}, {e^{λ1t}, e^{λ2t}}then is a basis for S.

Determine whether the set B is a basis for the vector space V.

If T : U → V and S : V → W are linear transformations such that range(T) ⊆ ker(S), what can be deduced about S ͦ T?

Determine whether the set B is a basis for the vector space V.

Determine whether the set B is a basis for the vector space V.

Determine whether the linear transformation T is (a) one-to-one and (b) onto.

T : P_{2} â†’ M_{22} defined by

Let T : R_{3}â†’ P_{2}be a linear transformation for which

and

Find

and

Test the sets of functions for linear independence in F. For those that are linearly dependent, express one of the functions as a linear combination of the others.

{sin x, sin 2x, sin 3x}

Find either the nullity or the rank of T and then use the Rank Theorem to find the other.

T : P_{2} → R defined by T(p)x)) = p′(0)

Test the sets of functions for linear independence in F. For those that are linearly dependent, express one of the functions as a linear combination of the others.

{1, ln(2x), ln(x^{2})}

Find the matrix [T]_{Câ†B}of the linear transformation T : V â†’ W with respect to the bases B and C of V and W, respectively. Verify Theorem 6.26 for the vector v by computing T(v) directly and using the theorem.

T : M_{22} â†’ M_{22} defined by T(A) = A - A^{T}, B = C = {E_{11},E_{12},E_{21},E_{22},},

Find the solution of the differential equation that satisfies the given boundary condition(s).

h'' - 4h' + 5h = 0, h(0) 0, h'(0) = -1

Find the matrix [T]_{Câ†B}of the linear transformation T : V â†’ W with respect to the bases B and C of V and W, respectively. Verify Theorem 6.26 for the vector v by computing T(v) directly and using the theorem.

T : M_{22} â†’ M_{22} defined by T(A) = AB - BA, where

B = C = {E_{11},E_{12},E_{21},E_{22},},

Determine whether T is a linear transformation.

T : F → F defined by T(f) = f(x))^{2}

Find the solution of the differential equation that satisfies the given boundary condition(s).

f '' - 2f ' + 5f = 0, f (0) = 1, f (π/4) = 0

Find the matrix [T]_{Câ†B}of the linear transformation T : V â†’ W with respect to the bases B and C of V and W, respectively. Verify Theorem 6.26 for the vector v by computing T(v) directly and using the theorem.

Repeat Exercise 9 with

B = {E_{22},E_{21},E_{12},E_{11},}, and C = {E_{12},E_{21},E_{22},E_{11},}.

**Data From Exercise 9**

T : M_{22} â†’ M_{22} defined by T(A) = A^{T}, B = C = {E_{11},E_{12},E_{21},E_{22},},

Find the solution of the differential equation that satisfies the given boundary condition(s).

y'' - 2ky' + k^{2}y = 0, k ≠ 0, y (0) = 1, y (1) = 0

Find the matrix [T]_{Câ†B}of the linear transformation T : V â†’ W with respect to the bases B and C of V and W, respectively. Verify Theorem 6.26 for the vector v by computing T(v) directly and using the theorem.

T : M_{22} â†’ M_{22} defined by T(A) = A^{T}, B = C = {E_{11},E_{12},E_{21},E_{22},},

Follow the instructions for Exercises 1–4 using p(x) instead of x.

p(x) = 1 + x^{2}, B = {1 + x + x^{2}, x + x^{2}, x^{2}}, C = {1, x, x^{2}} in P_{2}_{ }**Instructions**** From Exercise 1**

(a) Find the coordinate vectors [x]_{B} and [x]_{B} of x with respect to the bases B and C, respectively.

(b) Find the change-of-basis matrix P_{C}←_{B} from B to C.

(c) Use your answer to part (b) to compute and compare your answer with the one found in part (a).

(d) Find the change-of-basis matrix P_{B}←_{C }from C to B.

(e) Use your answers to parts (c) and (d) to compute [x]_{B}, and compare your answer with the one found in part (a).

Find the solution of the differential equation that satisfies the given boundary condition(s).

g'' - 2g = 0, g(0) = 1, g(1) = 0

Find the matrix [T]_{Câ†B}of the linear transformation T : V â†’ W with respect to the bases B and C of V and W, respectively. Verify Theorem 6.26 for the vector v by computing T(v) directly and using the theorem.

T : P_{2} â†’ R^{2} defined by T(P(x) )

B = {x^{2}, x, 1},

v = p(x) = a + bx + cx^{2}

Follow the instructions for Exercises 1–4 using p(x) instead of x.

p(x) = 3 + 2x, B = {1 + x, 1 - x}, C = {2x, 3} in P_{1 }**Instructions**** From Exercise 1**

(a) Find the coordinate vectors [x]_{B} and [x]_{B} of x with respect to the bases B and C, respectively.

(b) Find the change-of-basis matrix P_{C}←_{B} from B to C.

(c) Use your answer to part (b) to compute and compare your answer with the one found in part (a).

(d) Find the change-of-basis matrix P_{B}←_{C }from C to B.

(e) Use your answers to parts (c) and (d) to compute [x]_{B}, and compare your answer with the one found in part (a).

Determine whether the given set, together with the specified operations of addition and scalar multiplication, is a vector space. If it is not, list all of the axioms that fail to hold.

R^{2}, with the usual addition but scalar multiplication defined by

Find the solution of the differential equation that satisfies the given boundary condition(s).

x'' + x' - 12x = 0, x(0) = 0, x'(0) = 1

Test the sets of matrices for linear independence in M_{22}. For those that are linearly dependent, express one of the matrices as a linear combination of the others.

(a) Find the coordinate vectors [x]_{B}and [x]_{B}of x with respect to the bases B and C, respectively.

(b) Find the change-of-basis matrix P_{C}â†_{B}from B to C.

(c) Use your answer to part (b) to compute and compare your answer with the one found in part (a).

(d) Find the change-of-basis matrix P_{B}â†_{C }from C to B.

(e) Use your answers to parts (c) and (d) to compute [x]_{B}, and compare your answer with the one found in part (a).

(b) Find the change-of-basis matrix P

(c) Use your answer to part (b) to compute and compare your answer with the one found in part (a).

(d) Find the change-of-basis matrix P

(e) Use your answers to parts (c) and (d) to compute [x]

(b) Find the change-of-basis matrix P

(c) Use your answer to part (b) to compute and compare your answer with the one found in part (a).

(d) Find the change-of-basis matrix P

(e) Use your answers to parts (c) and (d) to compute [x]

(b) Find the change-of-basis matrix P

(c) Use your answer to part (b) to compute and compare your answer with the one found in part (a).

(d) Find the change-of-basis matrix P

(e) Use your answers to parts (c) and (d) to compute [x]

Let be defined by T : M_{22}â†’ R be defined by T(A) = tr (A).

(a) Which, if any, of the following matrices are in ker(T)?

(i)

(ii)

(iii)

(b) Which, if any, of the following scalars are in range(T)?

(i) 0

(ii) 5

(iii) -âˆš2

(c) Describe ker(T) and range(T).

Compute the pseudo inverse A^{+}of A in the given exercise.

Exercise 3

**Data From Exercise 3**

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.

Find cond_{1}(A) and cond_{âˆž }(A). State whether the given matrix is ill-conditioned.

Construct a linear (n, k, d) code or prove that no such code exists.

n = 8, k = 4, d = 4

Compute (a) ||A||_{2 }and (b) cond_{2}(A) for the indicated matrix.

Apply the Gram-Schmidt Process to the basis B to obtain an orthogonal basis for the inner product space V relative to the given inner product.

V = P_{2} [0,1], B = {1, 1 + x, 1 + x + x^{2}}, with the inner product in Example 7.5.

Find cond_{1}(A) and cond_{âˆž }(A). State whether the given matrix is ill-conditioned.

Construct a linear (n, k, d) code or prove that no such code exists.

n = 8, k = 5, d = 5

Compute (a) ||A||_{2 }and (b) cond_{2}(A) for the indicated matrix.

Find cond_{1}(A) and cond_{âˆž }(A). State whether the given matrix is ill-conditioned.

Construct a linear (n, k, d) code or prove that no such code exists.

n = 8, k = 2, d = 8

Compute (a) ||A||_{2 }and (b) cond_{2}(A) for the indicated matrix.

A in Exercise 8**Data From Exercise 8**

Construct a linear (n, k, d) code or prove that no such code exists.

n = 8, k = 1, d = 8

Compute (a) ||A||_{2 }and (b) cond_{2}(A) for the indicated matrix.

A in Exercise 3**Data From Exercise 3**

Compute the minimum distance of the code C and decode the vectors u, v, and w using nearest neighbor decoding.

C has generator matrix

(u,v) is an inner product. prove that the given statement is an identity

Prove that d(u, v) = √||u||^{2} + ||v||^{2 }if and only if u and v are orthogonal.

Compute the minimum distance of the code C and decode the vectors u, v, and w using nearest neighbor decoding.

In exercises (u,v) is an inner product. prove that the given statement is an identity

Prove that ||u + v || = ||u - v|| if and only if u and v are orthogonal.

Find the minimum distance of the codes.

The code with parity check matrix

Let ||A|| be a matrix norm that is compatible with a vector norm ||x||. Prove that ||A|| ≥ |λ| for every eigenvalue λ of A.

In exercises (u,v) is an inner product. prove that the given statement is an identity

{u, v} = 1/4 ||u + v||^{2} - 1/4|| u - v||^{2}

Find the minimum distance of the codes.

The code with parity check matrix P = [I | A], where

Find the minimum distance of the codes.

The n-times repetition code Rep_{n}.

Find the minimum distance of the codes.

The even parity code E_{n}.

Find vectors x and y with ||x||_{s}= 1 and ||y||_{m}= 1 such that ||A||_{1}= ||Ax||_{s}and ||A||_{âˆž}= ||Ay||_{m}, where A is the matrix in the given exercise.

Exercise 25

**Data From Exercise 25**

Find the minimum distance of the codes.

Find vectors x and y with ||x||_{s}= 1 and ||y||_{m}= 1 such that ||A||_{1}= ||Ax||_{s}and ||A||_{âˆž}= ||Ay||_{m}, where A is the matrix in the given exercise.

Exercise 24

**Data From Exercise 24**

Find the minimum distance of the codes.

Find vectors x and y with ||x||_{s}= 1 and ||y||_{m}= 1 such that ||A||_{1}= ||Ax||_{s}and ||A||_{âˆž}= ||Ay||_{m}, where A is the matrix in the given exercise.

Exercise 23

**Data From Exercise 23**

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|| = 2

Evaluate the expressions

||2u - 3v + w||

Find vectors x and y with ||x||_{s}= 1 and ||y||_{m}= 1 such that ||A||_{1}= ||Ax||_{s}and ||A||_{âˆž}= ||Ay||_{m}, where A is the matrix in the given exercise.

Exercise 22

**Data From Exercise 22**

Find vectors x and y with ||x||_{s}= 1 and ||y||_{m}= 1 such that ||A||_{1}= ||Ax||_{s}and ||A||_{âˆž}= ||Ay||_{m}, where A is the matrix in the given exercise.

Exercise 21

**Data From Exercise 21**

Find the best approximation to a solution of the given system of equations.

2x + 3y + z = 21

x + y + z = 7

-x + y - z = 14

-2y + z = 0

Find the Fourier coefficients a_{0}, a_{k}, and b_{k} of f on [-π, π].

f(x) = |x|

Find vectors x and y with ||x||_{s}= 1 and ||y||_{m}= 1 such that ||A||_{1}= ||Ax||_{s}and ||A||_{âˆž}= ||Ay||_{m}, where A is the matrix in the given exercise.

Exercise 20

**Data From Exercise 20**

Find the best approximation to a solution of the given system of equations.

x + y - z = 2

-y + 2z = 6

3x + 2y - z = 11

-x + z = 0

Compute ||A||_{F}, ||A||_{1}, and ||A||_{âˆž}.

Show that the least squares solution of Ax = b is not unique and solve the normal equations to find all the least squares solutions.

Find the outer product form of the SVD for the matrix in the given exercises.

Exercises 9 and 19

Exercises 9 and 19

**Data From Exercise 9 and 19**

Compute ||A||_{F}, ||A||_{1}, and ||A||_{âˆž}.

Find the outer product form of the SVD for the matrix in the given exercises.

Exercise 7 and 17

Exercise 7 and 17

**Data From Exercise 7 and 17**

Compute ||A||_{F}, ||A||_{1}, and ||A||_{âˆž}.

Find a least squares solution of Ax = b by constructing and solving the normal equations.

Find the outer product form of the SVD for the matrix in the given exercises.

Exercise 14

Exercise 14

**Data From Exercise 14**

Compute ||A||_{F}, ||A||_{1}, and ||A||_{âˆž}.

Find a least squares solution of Ax = b by constructing and solving the normal equations.

Find the outer product form of the SVD for the matrix in the given exercises.

Exercises 3 and 11

Exercises 3 and 11

**Data From Exercise 3 and 11**

Compute ||A||_{F}, ||A||_{1}, and ||A||_{âˆž}.

Show that ||1||^{2} = 2π and ||cos kx||^{2} = π in e[- π,π].

Find an SVD of the indicated matrix.

Compute ||A||_{F}, ||A||_{1}, and ||A||_{âˆž}.

Find an SVD of the indicated matrix.

A in Exercise 9

**Data From Exercise 9**

Prove Theorem 7.5(b).

Join SolutionInn Study Help for

1 Million+ Textbook Solutions

Learn the step-by-step answers to your textbook problems, just enter our Solution Library containing more than 1 Million+ textbooks solutions and help guides from over 1300 courses.

24/7 Online Tutors

Tune up your concepts by asking our tutors any time around the clock and get prompt responses.