Forming a Basis in R^3 - Explain the Correct Solution

[Chadlee88]

i have a question I am trying to work but I am not sure how to do it. I'm given 4 dirrerent answers to choose from (i won't post them because i want to try them myself)

Only one of the following 4 sets of vectors forms a basis of R3.
Explain which one is, and why, and explain why each of the other sets do not form a
basis.


S = {(1,1,1), (-2,1,1), (-1,2,2)}

This one is not because it cannot be expressed as a linear combination right??

 

[ircdan]

S is not a basis for R^3 because it is not linearly independent.

 

[HallsofIvy]

i have a question I am trying to work but I am not sure how to do it. I'm given 4 dirrerent answers to choose from (i won't post them because i want to try them myself)

Only one of the following 4 sets of vectors forms a basis of R3.
Explain which one is, and why, and explain why each of the other sets do not form a
basis.


S = {(1,1,1), (-2,1,1), (-1,2,2)}

This one is not because it cannot be expressed as a linear combination right??

Because what "cannot be expressed as a linear combination"?
Grammatically, the "it" in your sentence must refer to "this one", meaning the set of vectors- but it doesn't make sense to talk about expressing a set of vectors as a linear combination of anything.

It is true that S is not a basis for R³ because one of the vectors in S can be expressed as a linear combination of the other two. For example, (1, 1, 1)= -1(-2, 1, 1)+ 1(-1, 2, 2).