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
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 $\{ v_1, (0,1) \}$.