Number of vectors needed to Span R^n

[samona]

I am reading my linear algebra book, and in the chapter on Spanning, I got the impression that for a set to span R^n, it must contain at least n vectors. I confirmed that by searching through the forums.

However, I've reached the chapter on Linear Independence and in one of the examples it shows that a set containing 2 vectors can span R^3. So now I'm totally confused. Any help will be appreciated.

Example in the book:

S1= {[1,0,1], [0,1,1]}, S2 = {[1,0,1], [0,1,1], [3,2,5]}. Book says S1 and S2 span R^3, and prefers S1 since its "more efficient".

 

[lurflurf]

S1 and S2 have the same span, but it is not R3. For example [1,1,0] is not in their span.

 

[samona]

Thanks for the help. I just read this page like 100 times, and after reading your answer I read it again and missed an important detail.

The book is describing a subset of R^3 where W = [a, b, a+b] is a subspace of R^3. I still don't understand why they claim that S1 spans W though.

 

[lurflurf]

It does
W=span(S1)=Span(S2)
what is
a*[1,0,1]+b*[0,1,1]?

 

[samona]

that equals [a, b, a+b] which is in W. I see that it works, but what's confusing me is the R^3. Even though W is a subspace of R^3 shouldn't W need 3 vectors in order to be spanned? How come 2 vectors are able to span it when its in R^3?

I wish they left out the R^3 and just put down R^2, then it would make sense to me. Maybe I'll edit that page in my book lol

Thanks for your patience. I'm sure this is something that should be simple to understand.

 

[lurflurf]

Our vectors are in R3. A subspace need not span the space

{[1,2,3,4,5,6,7,0,0,0,0,0,0,0,0,0,0,0,0,0],[1,2,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]}
is a subspace of R20, but only has dimension 2 like W

 

[Fredrik]

that equals [a, b, a+b] which is in W. I see that it works, but what's confusing me is the R^3. Even though W is a subspace of R^3 shouldn't W need 3 vectors in order to be spanned? How come 2 vectors are able to span it when its in R^3?

I wish they left out the R^3 and just put down R^2, then it would make sense to me. Maybe I'll edit that page in my book lol

Thanks for your patience. I'm sure this is something that should be simple to understand.

Looks like you need to look at the definition of "subspace" again. A subspace is just a subset that's also a vector space. Your W is a subspace of $\mathbb R^3$ because a) it's a vector space, and b) it's a subset of $\mathbb R^3$.

The set $\{(x,y,z)\in\mathbb R^3|y=z=0\}$ ("the x axis") is a subspace of $\mathbb R^3$ that's spanned by a set containing only one vector: $\{(1,0,0)\}$.

 

[samona]

Our vectors are in R3. A subspace need not span the space

{[1,2,3,4,5,6,7,0,0,0,0,0,0,0,0,0,0,0,0,0],[1,2,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]}
is a subspace of R20, but only has dimension 2 like W

Got it! I see what you mean. That example actually clicked...finally. Thanks so much!

 

[samona]

Looks like you need to look at the definition of "subspace" again. A subspace is just a subset that's also a vector space. Your W is a subspace of $\mathbb R^3$ because a) it's a vector space, and b) it's a subset of $\mathbb R^3$.

The set $\{(x,y,z)\in\mathbb R^3|y=z=0\}$ ("the x axis") is a subspace of $\mathbb R^3$ that's spanned by a set containing only one vector: $\{(1,0,0)\}$.

Did look over it again. Your example really helped clear things up some more. Thank you so much!