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hello I am now a researcher in wireless communication, and now I need to use vector space with Matlab, but my knowledge is little in vector space so I have the first question given the following vector how I generate the span of this vector over the field $F$. Example :

$F_3=\{0,1,2\}$, and vectors $v_1=(1,0)$,$v_2=(2,2)$ ,$v_3=(1,2)$. From the definition of $span(v_1,v_2,v_3)=a_1v_1+a_3v_3+a_3v_3$. so the span contain the vectors :

$(0v_1+0v_2+0v_3),(0v_1+0v_2+1v_3),(0v_1+0v_2+2v_3),(0v_1+1v_2+0v_3),(0v_1+1v_2+1v_3),(0v_1+1v_2+2v_3),(0v_1+2v_2+0v_3),(0v_1+2v_2+1v_3),(0v_1+2v_2+2v_3),(1v_1+0v_2+0v_3),(1v_1+0v_2+1v_3),(1v_1+0v_2+2v_3),(1v_1+1v_2+0v_3),(1v_1+1v_2+1v_3),(1v_1+1v_2+2v_3),(1v_1+2v_2+0v_3),(1v_1+2v_2+1v_3),(1v_1+2v_2+2v_3),(2v_1+0v_2+0v_3),(2v_1+0v_2+1v_3),(2v_1+0v_2+2v_3),(2v_1+1v_2+0v_3),(2v_1+1v_2+1v_3),(2v_1+1v_2+2v_3),(2v_1+2v_2+0v_3),(2v_1+2v_2+1v_3),(2v_1+2v_2+2v_3)$

am I correct or not i how i found the basis of this span An the equation with size of fiel and nbr of vector to get number of element in the span

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  • $\begingroup$ You can put the vectors $v_1,v_2,v_3$ as rows of a matrix. Then make it do row reduction until it is in echelon form using only row transformations. The non-zero rows form a basis of the span. However, Matlab is not well suited for working over finite fields. I would use GAP instead. $\endgroup$ – Hellen Jul 22 '17 at 14:58
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You have found the span correctly, but the question is can we further simplify it.

Notice that $$2v_1+v_2 = v_3$$

Hence $v_3$ can be generated using $v_1$ and $v_2$.

Check that $$c_1v_1+c_2v_2=(0,0)$$ only has the trivial solution and hence $\{ v_1, v_2\}$ forms a basis.

A general element in the span can be written in the form of $d_1v_1+d_2v_2$. There are a total of $9$ elements which are just $F_3^2$. Another choice of basis could be $\{ v_1, (0,1) \}$.

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  • $\begingroup$ ok thank this my fist question i will have another question $\endgroup$ – Mokh Tar Bou Jul 22 '17 at 15:04

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