What is the dimension of S=span{v1,v2,v3} a subset of R3, where v1=(1 0 1) v2=(1 1 0)
v3=(1 -1 2) ? If the dimension is less than three, find a subset {v1,v2,v3} that is a basis for S and expand this basis to a basis for R3.

(I'm fairly certain the dimension is less than 3, as 2v1-v2-v3=0, which would imply that this is not linearly independent. I'm just not sure how to find a basis from here.

Question

What is the dimension of S=span{v1,v2,v3} a subset of R3, where v1=(1 0 1)  v2=(1 1 0)   
v3=(1 -1 2) ? If the dimension is less than three, find a subset {v1,v2,v3} that is a basis for S and expand this basis to a basis for R3. 

(I'm fairly certain the dimension is less than 3, as 2v1-v2-v3=0, which would imply that this is not linearly independent. I'm just not sure how to find a basis from here.

zzr0ck3r · Answer

im pretty sure you row reduce and then the columns with a leading 1 can be the basis

anonymous · Answer

Okay, thank you. I placed the vectors in a matrix 
A=1 0 1 0
     0 1 -1 0

anonymous · Answer

A= 1 1 1 0
      0 1 -1 0
      1 0 2 0
and row reduced from here. To find the basis, I only take the columns leading with 1 and leave the others out?

zzr0ck3r · Answer

im pretty sure, I was hoping someone else would say something...its been a bit since lin alg..

zzr0ck3r · Answer

you didnt need 0 0 0

anonymous · Answer

Okay, thank you. I had a general idea and was trying to replicate a similar example, but didn't think it was correct.

zzr0ck3r · Answer

does this look familiar with the other example? or can you tell if I was right?

zzr0ck3r · Answer

and to extend it to span R^3 we just add a vector such that the set of vectors are lin ind

zzr0ck3r · Answer

because we need 3 lin ind vectors to span R^3

anonymous · Answer

It looks similar to the example, and I think it is correct, because I have two linearly independent vectrs, v1=(1 0  0) and v2=(0 1 0) which makes the choice for the third v3=(0 0 1)

zzr0ck3r · Answer

nono when I say we take the two column vectors I mean the original ones that are in the spots of the reduced form

zzr0ck3r · Answer

A= 1 1 1 0 
     0 1 -1 0 
     1 0 2 0
what was the rref ?

anonymous · Answer

A= 1 0 2
      0 1 -1
      0 0 0

zzr0ck3r · Answer

so take
1 0 1 and 1 1 0

zzr0ck3r · Answer

and now notice
2v1-v2 = v3

anonymous · Answer

Oh, okay. The original vectors from those columns.

zzr0ck3r · Answer

i guess we could have just used the fact that 2v1-v2-v3=0
so
2v1-v2=v3
and thus v3 is repetitive...but this gives me hope that my method was right as well:)

anonymous · Answer

Oh, okay. I never thought to remove v3. Thank you, I'm pretty sure this is correct! :)