spanning?

[philipc]

I was just wondering if I can make this statment, if I was to prove any three vectors of R^3 are linear independent, can I also say those three vectors span R^3?

Philip

[chroot]

No; a set of vectors can be linearly independent without spanning a space, and a set of vectors can span a space without being linearly independent. A set of vectors that is both linearly independent and spans a space is a basis for that space.

- Warren

[enigma]

Yes Warren, but it isn't possible to have more than three linearly independant vectors in R3, therefore any set of three will span and be a basis.

I think you just lost the forest for the trees :wink:

[philipc]

Ok thanks for your help, can I ask one more, well maybe only one :biggrin:

I'm looking at my notes and kind of stuck with something,

Lets say I have the vectors [1,0] and [-1,1] and I'm asked what is the span?
so I have a[1,0] +b[-1,0] = [a-b,b] then it says this spans R^2 because [x,y] = [(x+y)-y,y] so x+y=a and y=b

not sure if I was sleeping during class and wrote the notes down wrong or what, but I'm not sure where the [x,y] = [(x+y)-y,y] came from?
Thanks again for your help
Philip

[matt grime]

You've shown how to write any vector in R^2 as a combination of [1,0] and [-1,1], therefore is spans. [x,y] = (x+y)[1,0] + y[-1,1]

[philipc]

but it was not the work of me?

so is that saying I need to find values for a and b to make it = [x,y], and that can be any combo of x and y for the values of a and b? Sorry to sound so dumb, but not really sure if I'm getting it.
Philip

[matt grime]

yep, you need to solve the vector equations a[1,0]+b[-1,1] = [x,y]

or splitting it into components

a-b=x
b=y

ie b=y, and a-y=x, a=x+y

[philipc]

OK now I see where it came from, thanks for the help.
Philip

[HallsofIvy]

Any basis for a vector space has three properties:

1. It spans the space
2. The vectors in it are linearly independent
3. The number of vectors in it is the same as the dimension of the space.

If any two of those are true, then the third is also true.

If you know that a set of three vectors in R3 is independent, then the must span the space.

(Strictly speaking, we define a basis as a set of vectors satisfying (1) and (2), show that any basis must have the same number of vectors and then define the dimension of the space to be that number.)