What the meaning of span space?

1. Sep 13, 2011

neilqiqi

1. The question goes that "Find two orthogonal vectors that span the same space as the two vectors"
The thing is, I know the definition of SPAN, say, a vector b is in span{v1, v2......vp} if x1v1+x2v2+....+xpvp = b. But what's the meaning of "span the same space"?

2. Relevant equations

3. The attempt at a solution

2. Sep 13, 2011

Tomer

if two sets of vectors "span the same space", it means that the new sets, spanned by these sets, are equal.
For example, sp {(1,0), (0,1)} = sp {(1,1), (1,0)}, while both of the spans are the set R^2.

In general, when you get two sets and you want to prove that they span the same space, you need to show that every element in the first span is also an element in the second span, and vice versa.
Hope it's clearer...