From a basis, find a cartesian linear system of equation

[God]

Hi,

So the basis is ((1,0,-1,0,1,0) , (-1,1,0,0,0,1) , (0,-1,1,1,0,0) , (0,0,0,1,0,0) , (0,0,0,0,1,0)), and I have to find a cartesian system of equation describing this set

I thought doing it as I used to do it would work, but in this case it seems not to work :

x1*(1,0,-1,0,1,0)+x6* (-1,1,0,0,0,1)+x3* (0,-1,1,1,0,0)+x4*(0,0,0,1,0,0)+x5* (0,0,0,0,1,0)=(x1-x6 , x6-x3 , x3-x1 , x3+x4 , x5+x1 , x6)=(x,y,z,t,u,v)

then trying to express x,y,z,t,u,v in terms of x,y,z,t,u,v (getting rid of the xi's), of course we have v=x6 so I can replace all x6's by v,
but I realised while trying to do it that I can't get rid of x4... since it appears only one time... we have no info on it. So I can't express t without x4... Same for x5. I tried to set x5=0 and x4=0 (only way to solve this...) but of course it gave me wrong equations at the end... (I tried with one of the vector it didn't always work)


so... how to proceed ? thanks in advance

 

[God]

tl;dr version :

basis is ((1,0,-1,0,1,0) , (-1,1,0,0,0,1) , (0,-1,1,1,0,0) , (0,0,0,1,0,0) , (0,0,0,0,1,0))
how to find a system of linear equations describing the space it creates ? pretty please

 

[HallsofIvy]

tl;dr version :

basis is ((1,0,-1,0,1,0) , (-1,1,0,0,0,1) , (0,-1,1,1,0,0) , (0,0,0,1,0,0) , (0,0,0,0,1,0))
how to find a system of linear equations describing the space it creates ? pretty please

The "space it creates" is the span of the set of vectors. In particular, any vector in the span of these is f the form
a(1, 0, -1, 0, 1, 0)+ b(-1, 1, 0, 0, 0, 1)+ c(0, -1, 1, 1, 0, 0)+ d(0, 0, 0, 1, 0)+ e(0, 0, 0, 0, 1, 0)= (a- b, b+ c, -a+ c, a+ d+ e, b). I'm not sure what "system of linear equations" you are referring to, but if (u, v, w, x, y, z) is a vector in that span, then we must have (u, v, w, x, y, z)= (a- b, b+ c, -a+ c, a+ d+ e, b) which gives the system of equations a- b= u, b+ c= v, -a+ c= w, a+ d+ e= y, b= z.

 

[God]

Sorry, I'm foreign so I don't know how to express myself, it's too technical and these words weren't taught to me in English class.
So I'll just use an example

((1,0,-1,0,1,0);(-1,1,0,0,0,1);(0,-1,1,1,0,0)) is a basis

a*(1,0,-1,0,1,0)+b*(-1,1,0,0,0,1)+c*(0,-1,1,1,0,0)=(a-b,b-c,c-a,c,a,b)=(x,y,z,t,u,v)
so the system I'm looking for is {x=u-v, y=v-t, z=t-u}

now the basis is ((1,0,-1,0,1,0) , (-1,1,0,0,0,1) , (0,-1,1,1,0,0) , (0,0,0,1,0,0) , (0,0,0,0,1,0))

a*(1,0,-1,0,1,0) +b* (-1,1,0,0,0,1) +c* (0,-1,1,1,0,0) +d* (0,0,0,1,0,0) +e* (0,0,0,0,1,0)=(a-b,b-c,c-a,c+d,a+e,b)=(x,y,z,t,u,v)

how can I find the system now ? with x,y,z,t,u,v
"a,b,c,d,e,f" shouldn't be used in the system - only x,y,z,t,u,v

somebody told me it the system for this one should be only one equation, but I still don't get it...

 

[HallsofIvy]

We still don't know what you mean by "a system of equations describing this set".