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
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?
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
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.
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.I don't see how this extra vector affects it though?