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Chapter 5: Q4E (page 215)
Find the angle between each of the pairs of vectors and in exercises 4 through 6.
4.
The angle between and is about .
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Show that an orthogonal transformation Lfrom to preserves angles: The angle between two nonzero vectors andinequals the angle between and .Conversely, is any linear transformation that preserves angles orthogonal.
Let Abe the matrix of an orthogonal projection. Find in two ways:
a.Geometrically. (Consider what happens when you apply an orthogonal projection twice.)
b.By computation, using the formula given in Theorem 5.3.10
TRUE OR FALSE? The equation holds for all matrices A.
This exercise shows one way to define the quaternions,discovered in 1843 by the Irish mathematician Sir W.R. Hamilton (1805-1865).Consider the set H of all matrices M of the form
where p,q,r,s are arbitrary real numbers.We can write M more sufficiently in partitioned form as
where A and B are rotation-scaling matrices.
a.Show that H is closed under addition:If M and N are in H then so is
c.Parts (a) and (b) Show that H is a subspace of the linear space .Find a basis of H and thus determine the dimension of H.
d.Show that H is closed under multiplication If M and N are in H then so is MN.
e.Show that if M is in H,then so is .
f.For a matrix M in H compute .
g.Which matrices M in H are invertible.If a matrix M in H is invertible is necessarily in H as well?
h. If M and N are in H,does the equationalways hold?
Are the rows of an orthogonal matrix A necessarily orthonormal?
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