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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}

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