Confused with working out the span of a set of vectors in R^2

gilabert1985

Hi everyone!

I have the following problem which I don't understand... It is already solved, but there are three questions I have regarding it.

The problem says: "Let S be the set of all vectors [itex]x=(x_{1}, x_{2})[/itex] in [itex]\Re^{2}[/itex] such that [itex]x_{1}=1[/itex]. What is the span of S?"

And here is the answer that has me so confused...

"[itex]span S = \Re^{2}[/itex] because [itex](x_{1}, x_{2})=x_{1}(1, x^{-1}_{1}x_{2}[/itex] when [itex]x_{1}\neq0[/itex]
and [itex](x_{1}, x_{2})=(1, 0)-(1, -x_{2})[/itex] when [itex]x_{1}=0[/itex]."

But I don't understand the first line... why does it say when [itex]x_{1}\neq0[/itex] if [itex]x_{1}[/itex] is supposed to be equal to 1?

And in the second line, the same... why is [itex]x_{1}=0[/itex]?

So yeah, I understand they are linear combinations and all that, but for the condition given ([itex]x_{1}=1[/itex]), I don't understand how this answer satisfies it.

Office_Shredder

The points (x₁,x₂) in the lines that are confusing you are not points in S, but points in R², which it then shows how to represent as linear combinations of points in S

Fredrik

Since you can't rewrite ##x_2## as ##x_1 x_1{}^{-1}x_2## when ##x_1=0##, you have to consider the case ##x_1=0## separately.