Reverses and Exercises: Difference between pages

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''Reverses'' are unary operations in geometric algebra that are analogs of conjugate or transpose operations.
These are exercises accompanying the book [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''Projective Geometric Algebra Illuminated''].


For any element $$\mathbf u$$ that is the [[wedge product]] of $$k$$ vectors, the ''reverse'' of $$\mathbf u$$, which we denote by $$\mathbf{\tilde u}$$, is the result of multiplying those same $$k$$ vectors in reverse order. For example, the reverse of $$\mathbf e_{423}$$ is $$\mathbf e_3 \wedge \mathbf e_2 \wedge \mathbf e_4$$, which we would write as $$-\mathbf e_{423}$$since 324 is an odd permutation of 423. In general, the reverse of an element $$\mathbf u$$ is given by
== Exercises for Chapter 2 ==


:$$\mathbf{\tilde u} = (-1)^{\operatorname{gr}(\mathbf u)(\operatorname{gr}(\mathbf u) - 1)/2}\,\mathbf u$$ .
'''1.''' Show that Equation (2.35) properly constructs a line containing two points $$\mathbf p$$ and $$\mathbf q$$ with non-unit weights by considering $$\mathbf p / p_w \wedge \mathbf q / q_w$$ and then scaling by $$p_wq_w$$.


Symmetrically, for any element $$\mathbf u$$ that is the [[antiwedge product]] of $$m$$ antivectors, the ''antireverse'' of $$\mathbf u$$, which we denote by $$\smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}}$$, is the result of multiplying those same $$m$$ antivectors in reverse order (but this time under the [[antiwedge product]]). In general, the antireverse of an element $$\mathbf u$$ is given by
'''2.''' Let $$\mathbf u$$ be a basis element of the 4D projective algebra. Prove that if $$\mathbf u \wedge \overline{\mathbf u} = {\large\unicode{x1D7D9}}$$ and $$\underline{\mathbf u} \wedge \mathbf u = {\large\unicode{x1D7D9}}$$, then it must also be true that $$\mathbf u \vee \overline{\mathbf u} = \mathbf 1$$ and $$\underline{\mathbf u} \vee \mathbf u = \mathbf 1$$. That is, show that right and left complements under the wedge product are also the right and left complements under the antiwedge product.


:$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} = (-1)^{\operatorname{ag}(\mathbf u)(\operatorname{ag}(\mathbf u) - 1)/2}\,\mathbf u$$ .
'''3.''' Suppose that the 4D trivectors $$\mathbf g$$ and $$\mathbf h$$ represent parallel planes in 3D space. Show that the magnitude of the moment of $$\mathbf g \vee \mathbf h$$ is the distance between the planes multiplied by both their weights.


The reverse and antireverse of any element $$\mathbf u$$ are related by
'''4.''' Let $$\mathbf m$$ be a $$4 \times 4$$ matrix that performs a rotation about the $$z$$ axis in homogeneous coordinates. Calculate the $$16 \times 16$$ exomorphism matrix $$\mathbf M$$ corresponding to $$\mathbf m$$.


:$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} = (-1)^{\operatorname{gr}(\mathbf u)\operatorname{ag}(\mathbf u)}(-1)^{n(n-1)/2}\,\mathbf{\tilde u}$$ ,
'''5.''' Suppose that $$\mathbf G$$ is a metric exomorphism. Use the fact that $$\mathbf G$$ is an exomorphism to prove that the associated antimetric $$\mathbb G$$ must satisfy $$\mathbb G(\mathbf a \vee \mathbf b) = \mathbb G\mathbf a \vee \mathbb G\mathbf b$$ for any $$\mathbf a$$ and $$\mathbf b$$.


where $$n$$ is the number of dimensions in the algebra. To extend the reversals to all elements of an algebra, we simply require that it is a linear operation. For any basis elements $$\mathbf x$$ and $$\mathbf y$$, and for any scalars $$a$$ and $$b$$, we must have, for the reverse
'''6.''' Suppose that the metric tensor $$\mathfrak g$$ is invertible. Show that the wedge and antiwedge products satisfy the relationship $$\mathbf a \vee \mathbf b = (\mathbf a^\unicode["segoe ui symbol"]{x2605} \wedge \mathbf b^\unicode["segoe ui symbol"]{x2605})^\unicode["segoe ui symbol"]{x2606}$$.


:$$\widetilde{(a\mathbf x + b\mathbf y)} = a\mathbf{\tilde x} + b\mathbf{\tilde y}$$ ,
'''7.''' Suppose that $$\mathbf a$$ and $$\mathbf b$$ are basis elements of an $$n$$-dimensional exterior algebra and $$\operatorname{gr}(\mathbf a) + \operatorname{gr}(\mathbf b) = n$$. Show that $$(\mathbf a \wedge \mathbf b)^\unicode["segoe ui symbol"]{x2605} = \mathbf a^\unicode["segoe ui symbol"]{x2605} \mathbin{\unicode{x25CF}} \mathbf b$$.


and similarly for the antireverse.
'''8.''' Show that the geometric norm is idempotent. That is, show that $$\Vert \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}} \Vert = \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}}$$.


The following table lists the reverse and antireverse for all of the basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.
'''9.''' Derive the relationship between left and right interior products shown in Equation (2.110).


[[Image:Reverses.svg|720px]]
'''10.''' Derive Equation (2.159), which is the expansion analog of Equation (2.129).


== In the Book ==
'''11.''' Derive a formula for $$\mathbf u^{\unicode["segoe ui symbol"]{x2605}\unicode["segoe ui symbol"]{x2605}}$$, the double bulk dual of $$\mathbf u$$, that uses only $$\operatorname{gr}(\mathbf u)$$, $$\operatorname{ag}(\mathbf u)$$, and the determinant of the metric tensor $$\mathfrak g$$.
 
* Reverses and antireverses are introduced in Section 3.4.
 
== See Also ==
 
* [[Complements]]

Revision as of 04:46, 4 May 2024

These are exercises accompanying the book Projective Geometric Algebra Illuminated.

Exercises for Chapter 2

1. Show that Equation (2.35) properly constructs a line containing two points $$\mathbf p$$ and $$\mathbf q$$ with non-unit weights by considering $$\mathbf p / p_w \wedge \mathbf q / q_w$$ and then scaling by $$p_wq_w$$.

2. Let $$\mathbf u$$ be a basis element of the 4D projective algebra. Prove that if $$\mathbf u \wedge \overline{\mathbf u} = {\large\unicode{x1D7D9}}$$ and $$\underline{\mathbf u} \wedge \mathbf u = {\large\unicode{x1D7D9}}$$, then it must also be true that $$\mathbf u \vee \overline{\mathbf u} = \mathbf 1$$ and $$\underline{\mathbf u} \vee \mathbf u = \mathbf 1$$. That is, show that right and left complements under the wedge product are also the right and left complements under the antiwedge product.

3. Suppose that the 4D trivectors $$\mathbf g$$ and $$\mathbf h$$ represent parallel planes in 3D space. Show that the magnitude of the moment of $$\mathbf g \vee \mathbf h$$ is the distance between the planes multiplied by both their weights.

4. Let $$\mathbf m$$ be a $$4 \times 4$$ matrix that performs a rotation about the $$z$$ axis in homogeneous coordinates. Calculate the $$16 \times 16$$ exomorphism matrix $$\mathbf M$$ corresponding to $$\mathbf m$$.

5. Suppose that $$\mathbf G$$ is a metric exomorphism. Use the fact that $$\mathbf G$$ is an exomorphism to prove that the associated antimetric $$\mathbb G$$ must satisfy $$\mathbb G(\mathbf a \vee \mathbf b) = \mathbb G\mathbf a \vee \mathbb G\mathbf b$$ for any $$\mathbf a$$ and $$\mathbf b$$.

6. Suppose that the metric tensor $$\mathfrak g$$ is invertible. Show that the wedge and antiwedge products satisfy the relationship $$\mathbf a \vee \mathbf b = (\mathbf a^\unicode["segoe ui symbol"]{x2605} \wedge \mathbf b^\unicode["segoe ui symbol"]{x2605})^\unicode["segoe ui symbol"]{x2606}$$.

7. Suppose that $$\mathbf a$$ and $$\mathbf b$$ are basis elements of an $$n$$-dimensional exterior algebra and $$\operatorname{gr}(\mathbf a) + \operatorname{gr}(\mathbf b) = n$$. Show that $$(\mathbf a \wedge \mathbf b)^\unicode["segoe ui symbol"]{x2605} = \mathbf a^\unicode["segoe ui symbol"]{x2605} \mathbin{\unicode{x25CF}} \mathbf b$$.

8. Show that the geometric norm is idempotent. That is, show that $$\Vert \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}} \Vert = \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}}$$.

9. Derive the relationship between left and right interior products shown in Equation (2.110).

10. Derive Equation (2.159), which is the expansion analog of Equation (2.129).

11. Derive a formula for $$\mathbf u^{\unicode["segoe ui symbol"]{x2605}\unicode["segoe ui symbol"]{x2605}}$$, the double bulk dual of $$\mathbf u$$, that uses only $$\operatorname{gr}(\mathbf u)$$, $$\operatorname{ag}(\mathbf u)$$, and the determinant of the metric tensor $$\mathfrak g$$.

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