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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 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}}

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