Hamiltons quaternion rule states that ij=k and ji=-k. How can

In Exercises 3–12, determine whether each set equipped with the given operations is a vector space. For those that are not vector spaces identify the vector space axioms that fail. The set of all pairs of real numbers of the form (x, y), where , with the standard operations on .

Product of two cyclic groups is cyclic iff their orders are co-'
Say you have two groups ${\displaystyle G=⟨g⟩}$ with order n and ${\displaystyle H=⟨h⟩}$ with order m. Then the product ${\displaystyle G\times H}$ is a cyclic group if and only if ${\displaystyle gcd(n,m)=1}$

I am at a very initial stage of commutative algebra. I want to know whether the power of a prime ideal in a commutative ring is prime ideal or not?

use long division to determine if 2x-1 is a factor of ${\displaystyle p\left(x\right)=6x^3+7x^2-x-2}$

As you can see the solution to the system is the coordinates of the point where the two lines intersect. So, when solving linear systems with two variables we are really asking where the two lineas will intersect.

Let $a=(a1,a2,a3)$ be a fixed vector in ${\mathbb{R}}^3$. Define the cross product $a\times v$ of $a$ and another vector $v=(v_1,v_2,v_3)\in {\mathbb{R}}^3$ as
$a\times v=det\left[\begin{array}{ccc}{e}_1& e_2& e_3\\ a_1& a_2& a_3\\ v_1& v_2& v_3\end{array}\right]$
Define a function $T:{\mathbb{R}}^3\to {\mathbb{R}}^3$ by $T(v)=a\times v$ for $v\in {\mathbb{R}}^3$.
a) Show that $T$ is a matrix transformation and calculate its representing matrix $M$
b) Find $\ker (T)$ and interpret its answer geometrically
c) find $\mathrm{range}(T)$ and interpret its answer geometrically

Does a non-invertible matrix transformation "really" not have an inverse?