Can 2 Vectors Span R3? Let's Find Out!

[caffeine19]

Hey there I was just wondering? Can 2 vectors span R3?
let's say I have i and j vectors. What are the examples that show i and j are the basis of R3 and span R3?

 

[micromass]

No, the dimension of \mathbb{R}^3 is 3, and this means exactly that you need at least 3 vectors to span the set!

 

[caffeine19]

but I have a question that asked me:
consider the following vectors R3: u-(1,2,3) v-(-1,1,2).
describe those 2 vectors for which they are a basis.
give an example of another vectors besides u and v that belongs to the set for which u and v are a basis.
I don't think I clearly get this question

 

[micromass]

Well, your two vectors u and v are clearly not a basis for \mathbb{R}^3. But they are a basis for a vector space smaller then \mathbb{R}^3, namely the span of u and v. The question wants you to calculate the span and give an example of a vector in it...

 

[caffeine19]

I'm really sorry, but this confuses me even more. If u don't mind explaining further,.

 

[micromass]

I'm really sorry, but this confuses me even more. If u don't mind explaining further,.

Do you know what the span of a set is?

 

[caffeine19]

the set of all possible linear combinations. I can give definition easily. I tend to get confused where to start.

 

[caffeine19]

I get confused what to do first.
The operations. I know how to find determinant, how to do gauss and so on. but I tend to mix them up in questions. which to use

 

[micromass]

Well, now they want you to calculate the span of u and v. So, what does an element of the span of u and v look like? It's a linear combination of u and v! So, can you use this to describe the span?

 

[ideasrule]

the set of all possible linear combinations. I can give definition easily. I tend to get confused where to start.

Do you know that the set of all linear combinations is just au+bv?

 

[caffeine19]

so basically, :
x.u + y.v = (a,b,c)
abc can be any real number.
yeah? am i on the right track? I really need help, this my weakest topic.

 

[caffeine19]

after a few tryouts I managed to solve it. Thank you all.