Quaternionic vector space

Key takeaways

Quaternionic vector space belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Quaternionic vector space to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Quaternionic vector space from memory before moving on to harder problems.

Reference excerpt

In noncommutative algebra, a branch of mathematics, a quaternionic vector space is a module over the quaternions. Since the quaternion algebra is division ring, these modules are referred to as "vector spaces". However, the quaternion algebra is noncommutative so we must distinguish left and right vector spaces. In left vector spaces, linear compositions of vectors v {\displaystyle v} and w {\displaystyle w} have the form a v + b w {\displaystyle av+bw} where a {\displaystyle a} , b ∈ H {\displaystyle b\in H} . In right vector spaces, linear compositions of vectors v {\displaystyle v} and w {\displaystyle w} have the form v a + w b {\displaystyle va+wb} . Similar to vector spaces over a field, if a quaternionic vector space has finite dimension n {\displaystyle n} , then it is isomorphic to the direct sum H n {\displaystyle H^{n}} of n {\displaystyle n} copies of the quaternion algebra H {\displaystyle H} . In this case we can use a standard basis which has the form

e 1 = ( 1 , 0 , … , 0 ) {\displaystyle e_{1}=(1,0,\ldots ,0)}

… {\displaystyle \ldots }

e n = ( 0 , … , 0 , 1 ) {\displaystyle e_{n}=(0,\ldots ,0,1)}

In a left quaternionic vector space H n {\displaystyle H^{n}} we use componentwise sum of vectors and product of vectors over scalars

( p 1 , … , p n ) + ( r 1 , … , r n ) = ( p 1 + r 1 , … , p n + r n ) {\displaystyle (p_{1},\ldots ,p_{n})+(r_{1},\ldots ,r_{n})=(p_{1}+r_{1},\ldots ,p_{n}+r_{n})}

q ( r 1 , … , r n ) = ( q r 1 , … , q r n ) {\displaystyle q(r_{1},\ldots ,r_{n})=(qr_{1},\ldots ,qr_{n})}

In a right quaternionic vector space H n {\displaystyle H^{n}} we also use componentwise sum of vectors and product of vectors over scalars

( p 1 , … , p n ) + ( r 1 , … , r n ) = ( p 1 + r 1 , … , p n + r n ) {\displaystyle (p_{1},\ldots ,p_{n})+(r_{1},\ldots ,r_{n})=(p_{1}+r_{1},\ldots ,p_{n}+r_{n})}

( r 1 , … , r n ) q = ( r 1 q , … , r n q ) {\displaystyle (r_{1},\ldots ,r_{n})q=(r_{1}q,\ldots ,r_{n}q)}

See also Vector space General linear group Special linear group SL(n,H) Symplectic group

References Harvey, F. Reese (1990). Spinors and Calibrations. San Diego: Academic Press. ISBN 0-12-329650-1.

Worked examples

Example 1 — a first encounter with Quaternionic vector space

Start with the simplest possible case. Write down what Quaternionic vector space claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Quaternionic vector space before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Quaternionic vector space ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Quaternionic vector space

In research: Quaternionic vector space appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Quaternionic vector space in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Quaternionic vector space is common in secondary-school and first-year university syllabi. It links to neighbouring topics Linear algebra, Linear algebra stubs, Quaternions, so understanding it makes those chapters shorter.
In everyday life: Look for Quaternionic vector space outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Quaternionic vector space in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Quaternionic vector space means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Quaternionic vector space out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Quaternionic vector space in simple terms?

In noncommutative algebra, a branch of mathematics, a quaternionic vector space is a module over the quaternions. Since the quaternion algebra is division ring, these modules are referred to as "vector spaces".

Why does Quaternionic vector space matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Quaternionic vector space?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Quaternionic vector space.