Finding the basis for a set of vectors.

[kingkong69]

The set is (1,3),(-1,2),(7,6)
it is in R2 so I don't get why there are 3 elements.
I assumed they are not vectors but points instead.


but if they are points then it becomes a line, and the answer is that its dimension is 2, and a basis is (1,0) and (0,1)

Could someone explain this?
Thanks

 

[DonAntonio]

The set is (1,3),(-1,2),(7,6)
it is in R2 so I don't get why there are 3 elements.
I assumed they are not vectors but points instead.


but if they are points then it becomes a line, and the answer is that its dimension is 2, and a basis is (1,0) and (0,1)

Could someone explain this?
Thanks

The question isn't clear: perhaps they meant to ask you to form a basis for $\mathbb R^2$ out of the three given vectors? If so the

answer is pretty simple: any two different vectors from your set of 3 forms a basis of that vector space, as none of them is

a scalar multiple of another one. Period

What you say about "points" and "vectors" and "lines" is odd, to say the least, and I can't tell what you actually meant.

DonAntonio

 

[HallsofIvy]

I strongly recommend you go back and read the problem again. You say titled this "Finding the basis for a set of vectors" which makes no sense. A general set of vectors does not have a basis! A vector space has a basis.

I suspect that the problem really says "find a subset of these vectors that form as basis for $R^2$". Since R² has dimension 2, as you say, a basis cannot have fewer than 2 vectors and giving you only two vectors would make the problem trivial- either just the 2 given vectors was a basis (if they are independent) or they are not. But if you are given three or more vectors, they cannot be independent. Any two independent vectors, chosen from those three, will be a basis for R².