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| Verify that Hamilton’s relations hold for the matrices 1, i, j, and k. Also show (assuming associativity and inverses) that these relations imply all the products of i, j, and k shown in Figure 1.2.

| Verify that Hamilton’s relations hold for the matrices 1, i, j, and k. Also show (assuming associativity and inverses) that these relations imply all the products of i, j, and k shown in Figure 1.2.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I dont know what this is asking abt quanternions for modern ALgebra

Ive attached notes, please explain. Thankyou.

 

Thus 1 behaves like the number 1, i? = -1 as before, and also j² = k? = -1. The noncommutativity is concentrated in the products of i, j, k, which are summarized in Figure 1.2. The product of any two distinct elements is k Figure 1.2: Products of the imaginary quaternion units. the third element in the circle, with a + sign if an arrow points from the first element to the second, and a sign otherwise. For example, ij = k, but ji = -k, so ij + ji. When Hamilton discovered H he described quaternion multiplication very con- cisely by the relations = j = k? = ijk =-1. 1.3.1 Verify that Hamilton’s relations hold for the matrices 1, i, j, and k. Also show (assuming associativity and inverses) that these relations imply all the products of i, j, and k shown in Figure 1.2.
Transcribed Image Text:Thus 1 behaves like the number 1, i? = -1 as before, and also j² = k? = -1. The noncommutativity is concentrated in the products of i, j, k, which are summarized in Figure 1.2. The product of any two distinct elements is k Figure 1.2: Products of the imaginary quaternion units. the third element in the circle, with a + sign if an arrow points from the first element to the second, and a sign otherwise. For example, ij = k, but ji = -k, so ij + ji. When Hamilton discovered H he described quaternion multiplication very con- cisely by the relations = j = k? = ijk =-1. 1.3.1 Verify that Hamilton’s relations hold for the matrices 1, i, j, and k. Also show (assuming associativity and inverses) that these relations imply all the products of i, j, and k shown in Figure 1.2.
1.3. Quaternions We saw that the complex numbers can be represented as 2 x 2 matrices. The quaternions extend the complex numbers. Quaternions were first described by Irish mathematician William Rowan Hamilton in 1843, and thus are denoted by H. Definition Н 3 {а + bi + cj+ dk | a, b, c, d € R}, where i, j, k are the basic quaternions subject the following multiplicative table : Quaternion multiplication table 1 i j k 1 1 ij i i -1 k -j k jj -k -1 i k k j -i -1 or the basic relations i? = j? = k² = ijk = -1. Note that quaternions are not commutative. 1.3. Quaternions The set of quaternions H forms a ring or a real vector space over basis {1, i, j, k}. Check this ! You have to check that multiplication is associative and distributive over addition. Just like we did with complex numbers, we can realize the quaternions as matrices, by defining the following matrices : 1 = I= J = ,K = Then check that these matrices satisfy the same group multiplication table as the basic quaternions above. And we denote -b – ic H = {al+bi+cJ+dk[a,b, c, d E R} = { a + id b— іс а — id 2, w E Now, verify that the map H → H given by al + bi + cj + dk → al+ b1 + cJ+ dK is an isomorphism of additive groups and that it commutes with the multiplication.
Transcribed Image Text:1.3. Quaternions We saw that the complex numbers can be represented as 2 x 2 matrices. The quaternions extend the complex numbers. Quaternions were first described by Irish mathematician William Rowan Hamilton in 1843, and thus are denoted by H. Definition Н 3 {а + bi + cj+ dk | a, b, c, d € R}, where i, j, k are the basic quaternions subject the following multiplicative table : Quaternion multiplication table 1 i j k 1 1 ij i i -1 k -j k jj -k -1 i k k j -i -1 or the basic relations i? = j? = k² = ijk = -1. Note that quaternions are not commutative. 1.3. Quaternions The set of quaternions H forms a ring or a real vector space over basis {1, i, j, k}. Check this ! You have to check that multiplication is associative and distributive over addition. Just like we did with complex numbers, we can realize the quaternions as matrices, by defining the following matrices : 1 = I= J = ,K = Then check that these matrices satisfy the same group multiplication table as the basic quaternions above. And we denote -b – ic H = {al+bi+cJ+dk[a,b, c, d E R} = { a + id b— іс а — id 2, w E Now, verify that the map H → H given by al + bi + cj + dk → al+ b1 + cJ+ dK is an isomorphism of additive groups and that it commutes with the multiplication.
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