Check the true statements below:

a. the orthogonal projection of yy onto vv is the same as the orthogonal projection of yy onto cvcv whenever c≠0c≠0.

b. if the columns of an m×nm×n matrix aa are orthonormal, then the linear mapping x→axx→ax preserves lengths.

c. if a set s={}s={} has the property that ui⋅uj=0ui⋅uj=0 whenever i≠ji≠j, then ss is an orthonormal set.

d. an orthogonal matrix is invertible.

e. not every orthogonal set in rnrn is a linearly independent set.

Asays "we are both knaves" and b says nothing. exercises 24–31 relate to inhabitants of an island on which there are three kinds of people: knights who always tell the truth, knaves who always lie, and spies (called normals by smullyan [sm78]) who can either lie or tell the truth. you encounter three people, a, b, and c. you know one of these people is a knight, one is a knave, and one is a spy. each of the three people knows the type of person each of other two is. for each of these situations, if possible, determine whether there is a unique solution and determine who the knave, knight, and spy are. when there is no unique solution, list all possible solutions or state that there are no solutions. 24. a says "c is the knave," b says, "a is the knight," and c says "i am the spy."

Find the exact value of each of the following. in each case, show your work and explain the steps you take to find the value. (a) sin 17π/6 (b) tan 13π/4 (c) sec 11π/3

Acourt reporter is transcribing an accident report from germany. the report states that the driver was traveling 80 kilometers per hour. how many miles per hour was the driver traveling?