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Chapter 5: Q18E (page 261)
Consider an orthonormal basis B of the inner product space V. For an element fof V, what is the relationship between and (the norm in defined by the dot product)?
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Find the length of each of the vectorsIn exercises 1 through 3.
If the matrices Aand Bare orthogonal, which of the matrices in Exercise 5 through 11 must be orthogonal as well?.
TRUE OR FALSE?If matrices A and Sare orthogonal, then is orthogonal as well.
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?
TRUE OR FALSE?If are two vectors in, then the equation role="math" localid="1659506190737" must hold.
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