The Integral Domain Hierarchy, Part 2. In any area of math, it's always good idea to keep a few counterexamples in your back pocket.
Integral Domain: a commutative ring with 1 where the product of any two nonzero elements is always nonzero
But if you haven't—and even if you have!—I hope you'll take a few minutes to enjoy it with me. Here's the idea:
A matrix is a two-dimensional array of numbers, while an n n -dimensional array of numbers is called a tensor of order n n or an ...
Such a mapping can be pictured as a node with two edges. One edge represents the input space, the other edge represents the output space.
If I◃R I ◃ R is an arbitrary nonzero ideal in the Euclidean domain R R , then I=(d) I = ( d ) , where d∈I d ∈ I such that d d has ...
This is consistent with the fact that the trace is a number. It's a 0-tensor so it has no free indices. Now here's a proof that the trace is invariant under ...
(So thus far, we've exploiting the fact that R3⊗R2 R 3 ⊗ R 2 is isomorphic to R6 R 6 .) You might refer to this as matrix-vector duality.
Here, U U and V V are unitary matrices, and hence isometries, and hence triangles. The matrix D D is a diagonal matrix, which I like to represent by a ...
A density matrix is a generalization of a unit vector—it accounts for interactions between the two particles.
This may seem to be in conflict with what we did above, but it's not! The two go hand-in-hand. Any m×n m × n matrix can be reshaped into a ...
where Tor(M)↪M Tor ( M ) ↪ M is inclusion and π:M→M/Tor(M) π : M → M / Tor ( M ) is the natural projection map. The exactness of the sequence is ...
An isometric embedding U U is a linear map from a space V V into a space W W of larger dimension that preserves the lengths of vectors.
But why the tensor product? Why is it that this construction—out of all things—describes the interactions within a quantum system so well, so naturally?
From our proposition above, it's clear that the Theorem is just one step away! One need only prove that
Mathematics Problem Book by Ildikó Reisch and László Szigeti, International Business School Budapest, 2001
Permutations Combinations Mathematical Induction and 15 7 Binomial Theorem 3. Sequence and Series 9 16
Last week we saw an example of how to use the rational canonical form (RCF) to classify matrices of a given order in GL2(Q) GL 2 ( Q ) .
Applications of Integrals : Application in finding the area under simple curves especially lines areas of
This string-with-a-bead has no free edges. It's a loop. This is consistent with the fact that the trace is a number. It's a 0-tensor so it has no free ...
I've only shared a snippet of basic arithmetic. For a deeper look into the mathematics, I recommend reading through Jeremy Kun's ...
We know what it means to have a module M M over a (commutative, say) ring R R . We also know that if our ring R R is actually a field, ...
Trigonometric functions Positive and negative angles Measurement of angles Seagesemal system and circular system Conversion from
to represent numbers and Express basic arithmetic processes algebraically Substitution Substitute numbers for letters in expressions
Here's a thought map which is (probably) more fun than practical. Note, p p and q q denote primes below:
Principles of Mathematical Analysis Differentiation And Integration, Mathematical Analysis, Real Number System, Advanced
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After some exposure to group theory, you quickly learn that when trying to prove a group G G is abelian, checking if xy=yx x y = y x for arbitrary ...
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... the following are official publications of Alcorn State University: THE ALUMNUS --- Official publication of the Alumni ASU TODAY Online Daily ...
Ordinary Differential Equations: Second & higher order linear differential equations with constant coefficients, General