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- Cardan angle (zxz, zxzxz) rotation - Mathematics Stack Exchange
On the wikipedia page there is a listing of 12 matrices that can be used to represent a yaw-pitch-roll rotation series (YXZ) as a ZXZ rotation, or an XZX rotation, or an XZY rotation
- Cardan (or Tait-Bryan) angles z-x-y and rotation matrix - Proof
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- Describe all group homomorphisms from Z×Z into Z
For any pair $(a,b)\in \mathbb Z\times\mathbb Z$ you find a group homomorphism by assigning $(x,y)\mapsto ax+by$
- How to define an injective and surjective function from
Since you are to show that $\mathbb{N}$ and $\mathbb{Z}$ have the same cardinality, you're correct: you need to find a bijection (hence both injective and surjective) between $\mathbb Z $ and $\mathbb N$
- abstract algebra - Applications of the Isomorphism theorems . . .
This is an application of the second isomorphism theorem, although the theorem does not play a crucial role in it
- reference request - What are usual notations for surjective, injective . . .
Update: In the category of sets, an epimorphism is a surjective map and a monomorphism is an injective map As is mentioned in the morphisms question, the usual notation is $\\rightarrowtail$ or $\\
- Find all subgroups of $\\mathbb{Z}\\times\\mathbb{Z}$.
By reason of comments underneath Makoto Koto's answer and spacing, I reworked the answer Let $\{(1, 0),(0, 1)\}$ be the canonical basis of $\mathbb{Z}\times\mathbb{Z}$
- abstract algebra - Describe all ring homomorphisms from $\mathbb Z . . .
Note: In this class, a ring homomorphism must map multiplicative and additive identities to multiplicative and additive identities This is different from our textbook's requirement, and often means
- Modules and epimorphisms - Mathematics Stack Exchange
Invariants of submodules of (bounded) torsion modules: If R is a PID and M is a torsion module, then M is the direct sum of its submodules M p = { m in M : there is an n such that p n ⋅m = 0 } where p varies over the set of (nonzero) primes
- rotation after translation as translation after rotation
Originally, I assumed you meant applying the operations as rotation-rotation-translation, but since you mean rotation-translation-rotation, my reasoning does not apply to your problem
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