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Chapter 5: Q47E (page 264)
TRUE OR FALSE
There exist a basis of that consists of orthogonal matrices.
The given statement is true.
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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?
Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.
5.
Let Abe anmatrix. Is the formulanecessarily true? Explain.
TRUE OR FALSE?If A and Bare symmetric matrices, then A+B must be symmetric as well.
If the nxnmatrices Aand Bare orthogonal, which of the matrices in Exercise 5 through 11 must be orthogonal as well? .
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