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The following is one way to define the quaternions, discovered in 1843 by the Irish mathematician Sir W. R. Hamilton. Consider the set H of all 4 x 4 matrices M of the form M = P 9 T S -q -T -S P S -T -8 P 9 T -9 P where p, q, r, s are arbitrary real numbers. We can write M more succinctly in partitioned form as M = A-B BAT 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 M + N. (b) Show that H is closed under scalar multiplication: If M is in H and k is an arbitrary scalar, then kM is in H. (c) The above show that H is a subspace of the linear space R4×4. Find a basis of H, and thus determine the dimension of H.

The following is one way to define the quaternions, discovered in 1843 by the Irish mathematician Sir W. R. Hamilton. Consider the set H of all 4 x 4 matrices M of the form M = P 9 T S -q -T -S P S -T -8 P 9 T -9 P where p, q, r, s are arbitrary real numbers. We can write M more succinctly in partitioned form as M = A-B BAT 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 M + N. (b) Show that H is closed under scalar multiplication: If M is in H and k is an arbitrary scalar, then kM is in H. (c) The above show that H is a subspace of the linear space R4×4. Find a basis of H, and thus determine the dimension of H.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.2: Mathematical Induction
Problem 30E
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The following is one way to define the quaternions, discovered in 1843 by the Irish mathematician Sir W. R. Hamilton. Consider the set H of all 4 x 4 matrices M of the form M = P 9 T S -q -T P S -S P 9 T -9 P -S -T where p, q, r, s are arbitrary real numbers. We can write M more succinctly in partitioned form as M = [ A ] A-B BAT 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 M + N. (b) Show that H is closed under scalar multiplication: If M is in H and k is an arbitrary scalar, then kM is in H. (c) The above show that H is a subspace of the linear space R4x4. 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 MT. (f) For a matrix M in H, compute MT M. (g) Which matrices M in H are invertible? If a matrix M in H is invertible, is M-¹ necessarily in H as well? (h) If M and N are in H, does the equation MN = NM always hold?
Transcribed Image Text:The following is one way to define the quaternions, discovered in 1843 by the Irish mathematician Sir W. R. Hamilton. Consider the set H of all 4 x 4 matrices M of the form M = P 9 T S -q -T P S -S P 9 T -9 P -S -T where p, q, r, s are arbitrary real numbers. We can write M more succinctly in partitioned form as M = [ A ] A-B BAT 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 M + N. (b) Show that H is closed under scalar multiplication: If M is in H and k is an arbitrary scalar, then kM is in H. (c) The above show that H is a subspace of the linear space R4x4. 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 MT. (f) For a matrix M in H, compute MT M. (g) Which matrices M in H are invertible? If a matrix M in H is invertible, is M-¹ necessarily in H as well? (h) If M and N are in H, does the equation MN = NM always hold?
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