Since quaternion algebra is division ring, then module over quaternion algebra is called vector space. Because quaternion algebra is non-commutative, we distinguish left and right vector spaces. In left vector space, linear composition of vectors v {\displaystyle v} and w {\displaystyle w} has form a v + b w {\displaystyle av+bw} where a {\displaystyle a} , b ∈ H {\displaystyle b\in H} . In right vector space, linear composition of vectors v {\displaystyle v} and w {\displaystyle w} has form v a + w b {\displaystyle va+wb} .

If quaternionic vector space has finite dimension n {\displaystyle n} , then it is isomorphic to direct sum H n {\displaystyle H^{n}} of n {\displaystyle n} copies of quaternion algebra H {\displaystyle H} . In such case we can use basis which has 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 left quaternionic vector space H n {\displaystyle H^{n}} we use componentwise sum of vectors and product of vector over scalar

( 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 right quaternionic vector space H n {\displaystyle H^{n}} we use componentwise sum of vectors and product of vector over scalar

( 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)}