# Spans

• Feb 5th 2009, 02:21 PM
sgc
Spans
Prove that span ({x})= {ax: a in F} for any vector x in a vector space. Interpret this result geometrically in R^3.

Now, I understand that the span of {x}= a1v1 + a2v2 + . . . anvn. But how does {ax} = a1v1+ a2v2+ . . . anvn? Unless I've got it all wrong.
• Feb 5th 2009, 02:53 PM
HallsofIvy
Quote:

Originally Posted by sgc
Prove that span ({x})= {ax: a in F} for any vector x in a vector space. Interpret this result geometrically in R^3.

Now, I understand that the span of {x}= a1v1 + a2v2 + . . . anvn.'

No, you don't understand that. That's not true because there is no "v1", "v2", ... "vn" in this problem. The span of a set of vectors is the set of all linear combinations of the vectors in that set. IF the set of vectors is {v1, v2, ..., v3} THEN the span is the set of vectors of the form a1v1+ a2v3+ ...+ anvn. But the span of a single vector {x} is the set of all multiples of x. What is that?

Quote:

But how does {ax} = a1v1+ a2v2+ . . . anvn? Unless I've got it all wrong.