# Linearility/spanning sets.

• Dec 9th 2010, 10:26 AM
chr91
Linearility/spanning sets.
Let {x1, x2... xn} be a spanning set for V.

If we add an additional vector xn+1 to the set , will we still have a spanning set?

I'm struggling to understand linear span. I know that if x1, x2.. xn is a spanning set for V, then it takes all the linear combinations of c1x1, c2x2.. cnxn.

I don't see how this extra vector affects it though?
• Dec 9th 2010, 01:49 PM
dwsmith
http://www.mathhelpforum.com/math-he...es-144330.html

2nd post proves both cases: n+1 is in the span and n+1 isn't in the span
• Dec 9th 2010, 02:43 PM
chr91
Quote:

Originally Posted by dwsmith
http://www.mathhelpforum.com/math-he...es-144330.html

2nd post proves both cases: n+1 is in the span and n+1 isn't in the span

Thanks alot!

You were of great help :)
• Dec 10th 2010, 03:25 AM
HallsofIvy
Quote:

Originally Posted by chr91
Let {x1, x2... xn} be a spanning set for V.

If we add an additional vector xn+1 to the set , will we still have a spanning set?

I'm struggling to understand linear span. I know that if x1, x2.. xn is a spanning set for V, then it takes all the linear combinations of c1x1, c2x2.. cnxn.

Phrased that way, it seems backwards. The point is not that all linear combinations are in V but that every vector in V is such a linear combination.

Quote:

I don't see how this extra vector affects it though?
It doesn't affect it! That's the whole point. If x1, x2, ..., xn is a spanning set for V, then any vector u in V can be written as a linear combination of them: u= a1x1+ a2x+ 2+ ...+ anxn, and any one of the "a"s can be 0. If put in an additional xn+1, just think of it as always having coefficient 0.