Proof that vectors span R3

[judahs_lion]

Show that vectors v1, v2, and v3 span R3.

V₁=(1,0,0)
V₂=(2,2,0)
V₃=(3,3,3)

I'm pretty sure I'm doing this wrong?

a(V₁) +b(V₂) +c(V₃) = [x,y,z]

for (a= 0, b = 0, c = 1/3)

[0,0,0] +[0,0,0] +[1,1,1] = [x,y,z]

[1,1,1] = [x,y,z]

 

[Mark44]

Show that vectors v1, v2, and v3 span R3.

V₁=(1,0,0)
V₂=(2,2,0)
V₃=(3,3,3)

I'm pretty sure I'm doing this wrong?

a(V₁) +b(V₂) +c(V₃) = [x,y,z]

for (a= 0, b = 0, c = 1/3)

[0,0,0] +[0,0,0] +[1,1,1] = [x,y,z]

[1,1,1] = [x,y,z]

Not right. In a nutshell you want to show that for an arbitrary vector <x, y, z>, there are some constants a, b, and c so that aV₁ +bV₂ +cV₃ = <x,y,z>.

You can do this by solving the matrix equation Ab = x for b, where the columns of matrix A are your vectors V₁, V₂, and V₃. The vector I show as b is <a, b, c>, and the vector I show as x is <x, y, z>.

 

[VeeEight]

Try showing that you can generate the standard basis of R³, {(1,0,0), (0,1,0), (0,0,1)}, using the elements v₁, v₂, v₃. For example, what combination of these vectors will give you (0,1,0)?

 

[judahs_lion]

Is there any way u dumb it down just a lil more , I still feel very lost.

 

[judahs_lion]

Ok, I got this far

SEE ATTACHMENT

 

[VeeEight]

You want to show that {v₁, v₂, v₃} = V spans R³. You already know that the vectors (1,0,0), (0,1,0), and (0,0,1) span R³. So you can try showing that V generates (0,1,0) and (0,0,1) and thus generates R³

 

[judahs_lion]

How do I know they span R3?

 

[Mark44]

Ok, I got this far

SEE ATTACHMENT

The work in the attachment looks fine. If you can row-reduce your matrix to the identity matrix [1 0 0; 0 1 0; 0 0 1], that's enough to guarantee that your three vectors span R³.