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Proof that vectors span R3

  • #1
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]
 

Attachments

Last edited:

Answers and Replies

  • #2
33,514
5,195
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>.
 
  • #3
614
0
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)?
 
  • #4
Is there any way u dumb it down just a lil more , I still feel very lost.
 
  • #5
Ok, I got this far

SEE ATTACHMENT
 

Attachments

  • #6
614
0
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
 
  • #7
How do I know they span R3?
 
  • #8
33,514
5,195
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.
 

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