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 $x=(x_{1}, x_{2})$ in $\Re^{2}$ such that $x_{1}=1$. What is the span of S?"

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

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

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

And in the second line, the same... why is $x_{1}=0$?

So yeah, I understand they are linear combinations and all that, but for the condition given ($x_{1}=1$), 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.