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Vector space is defined such as

VS = ( T , V , + , * )

Where is T is Division ring V is set of vectors and + and * are operations.

Having T defined such as

$T = (\{ 0, 1, 2 , 3 , 4 \} , + mod 5 , * mod 5 )$

and V set of vectors .

Can i understand it that vector space is set of all vectors that are created by

$\forall a \in T , \forall x \in V , y = a*x$

or better defined

for( a in T )     
     for ( x in A )
        y = x*a

The set of Y is the whole vector space?

2 Answers 2

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What you've written doesn't make sense to me, but I think you have an idea about what is happening.

The integers mod $5$, usually written $\mathbb{Z}_5$ (you call it $T$) is a field. For one example of a vector space the vectors $V$ might be the triples $(a,b,c)$ of elements of $\mathbb{Z}_5$. You can add those vectors and multiply them by scalars from $\mathbb{Z}_5$. This vector space is the analog of the real three dimensional vector space $\mathbb{R}^3$.

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    Thanks , i am using T cuz its the way we have it defined in our class. – 2017-02-25
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You have omitted some of the def'n. The operation $+$ is a binary operation on $V$ for which $(V,+)$ is a commutative group. The multiplication $*$ is between a member $t\in T$ and a member $v\in V$, with $t_1*v_1=v_1*t_1$, and with $t_1*(v_1+v_2)=(t_1*v_1)+(t_1*v_2)$ and with $t_1*(t_2*v_1)=(t_1t_2)*v_1$ for $t_1,t_2\in T$ and $v_1,v_2\in V$.

And $T$ must be a field (in English). The meaning of "field" differs in different countries or languages. But $T$ must be a commutative division ring with multiplicative unit. (That may be redundant.)

It is customary to omit the * in writing.