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I have the following matrix:

\begin{bmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}

So we have $v_2 = 0$ and $v_3 = 0$.

But my textbook claims that the span is $(1, 0, 0)$.

I don't understand where the $1$ is coming from? Shouldn't it just be $(0, 0, 0)$? If not why? I would greatly appreciate it if people could please take the time to explain this.

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    Do you have $x_1=0$?. since you don't the span should be ($x_1,0,0$) where $x_1$ is free to take any value. so you take for example $x_1=1$ or $2$ or anything – 2017-02-20
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    @Upstart thanks for the response. Can you please elaborate on this concept? – 2017-02-20
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    elaborate on what i guess i told you everything – 2017-02-20
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    @Upstart I mean give a more generalised and detailed explanation of the concept. – 2017-02-20
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    since the matrix corresponds to an infinite number of solutions to the system of equations.Where does that infiniteness come from? That comes from the fact that you cant find specific value for $x_1$ so you are free to choose any value for $x_1$ so choose $0$ or $1$ or $2$ or $3$ .but you gave a specific value ($0,0,0$) if you want to give all the solutions then just write span {($1,0,0$)}. span means that the scalar you have for the span includes $0 $ so you have you solution ($0,0,0$) as well and of course infinite others as well – 2017-02-20
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    @Upstart So having all 0s is the same as it being a free variable? – 2017-02-20
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    @ThePointer You said "so we have $v_2 = 0$ and $v_3 = 0$". It seems like you also want to say $v_1 = 0$, so that $(0,0,0)$ is the only solution. Why do you think we should be able to say that $v_1 = 0$? – 2017-02-20
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    @ThePointer your last question "So having all 0s is the same as it being a free variable?" is non-sensical to the point that it is not clear what you're *trying* to ask here. – 2017-02-20
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    @Omnomnomnom Upstart said that we are free to choose any value for the variable $x_1$ since we cannot find a specific value. This is what I understand to be known as a "free variable" of a system -- variables that have an infinite number of values. – 2017-02-20
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    @ThePointer okay, great. Yes: in this case, we would say that $x_1$ is a *free variable*, as you said. It's a variable that can be assigned any value and still yield a solution. – 2017-02-20
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    @ThePointer It seems to me that what's bothering you is something like this: *$v_1$ is a free variable, so we can assign it any value. Why would we choose to set $v_1 = 1$ as opposed to $v_1 = 0$? Why is it wrong to just take $v_1 = 0$?* Does that sound right? – 2017-02-20
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    @Omnomnomnom Yes, that was the original question. But now that I understand that a row of zeroes means that the leading variable is just a free variable, I understand the concept. – 2017-02-20
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    @ThePointer that was *not* your original question. It is, however, the meaning that I've extracted after reading your back and forth with Upstart. In the future, I hope that you'll be more precise in describing your issues. I'm glad that you feel you understand what's happening now. – 2017-02-20

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