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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 M22spanned by
Is M22 spanned by

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

V = P2, W = {p{x} in P3 : p(0) = 0}

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

V = P2, B = {2 + 3x - x2, 1 + 2x + 2x2}

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 = P2, B = {1 - x, 1 - x2, x - x2}

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 = P2, B = {x, 1 - x, 1 + x + x2}

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 Zp. If it is not, list all of the axioms that fail to hold.

Z6, over Z3 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 λ1 and λ2 be the roots of the characteristic equation λ2 + aλ +  b = 0.

b. If λ1 = λ2, {eλ1t, eλ2t}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 : P2 → M22 defined by 

Let T : R3→ P2be 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 : P2 → 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(x2)}

Find the matrix [T]C←Bof the linear transformation T : V → 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 : M22 → M22 defined by T(A) = A - AT, B = C = {E11,E12,E21,E22,}, 

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]C←Bof the linear transformation T : V → 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 : M22 → M22 defined by T(A) = AB - BA, where 

B = C = {E11,E12,E21,E22,}, 

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]C←Bof the linear transformation T : V → 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 = {E22,E21,E12,E11,}, and C = {E12,E21,E22,E11,}.


Data From Exercise 9

T : M22 → M22 defined by T(A) = AT, B = C = {E11,E12,E21,E22,},

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

y'' - 2ky' + k2y = 0, k ≠ 0, y (0) = 1, y (1) = 0

Find the matrix [T]C←Bof the linear transformation T : V → 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 : M22 → M22 defined by T(A) = AT, B = C = {E11,E12,E21,E22,}, 

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

p(x) = 1 + x2, B = {1 + x + x2, x + x2, x2}, C = {1, x, x2} in P2
 

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 PCB 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 PBfrom 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]C←Bof the linear transformation T : V → 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 : P2 → R2 defined by T(P(x) ) 

B = {x2, x, 1}, 

v = p(x) = a + bx + cx2

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 P1    

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 PCB 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 PBfrom 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.


R2, 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 M22. For those that are linearly dependent, express one of the matrices as a linear combination of the others.

(a) Find the coordinate vectors [x]Band [x]Bof x with respect to the bases B and C, respectively.
(b) Find the change-of-basis matrix PC←Bfrom 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 PB←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).

(a) Find the coordinate vectors [x]Band [x]Bof x with respect to the bases B and C, respectively.
(b) Find the change-of-basis matrix PC←Bfrom 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 PB←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).

Test the sets of matrices for linear independence in M22. For those that are linearly dependent, express one of the matrices as a linear combination of the others.

(a) Find the coordinate vectors [x]Band [x]Bof x with respect to the bases B and C, respectively.
(b) Find the change-of-basis matrix PC←Bfrom 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 PB←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).

Let be defined by T : M22→ 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) -√2

(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 cond1(A)  and cond∞ (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||and (b) cond2(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 = P2 [0,1], B = {1, 1 + x, 1 + x + x2}, with the inner product in Example 7.5.

Find cond1(A)  and cond∞ (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||and (b) cond2(A) for the indicated matrix.

Find cond1(A)  and cond∞ (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||and (b) cond2(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||and (b) cond2(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||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 Repn.

Find the minimum distance of the codes.

The even parity code En.

Find vectors x and y with ||x||s= 1 and ||y||m= 1 such that ||A||1= ||Ax||sand ||A||∞= ||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||sand ||A||∞= ||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||sand ||A||∞= ||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||sand ||A||∞= ||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||sand ||A||∞= ||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 a0, ak, and bk 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||sand ||A||∞= ||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||∞.

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


Data From Exercise 9 and 19

Compute ||A||F, ||A||1, and ||A||∞.

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.
Exercise 7 and 17


Data From Exercise 7 and 17

Compute ||A||F, ||A||1, and ||A||∞.

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


Data From Exercise 14

Compute ||A||F, ||A||1, and ||A||∞.

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


Data From Exercise 3 and 11

Compute ||A||F, ||A||1, and ||A||∞.

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||∞.

Find an SVD of the indicated matrix.

A in Exercise 9


Data From Exercise 9

Prove Theorem 7.5(b).

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