How does this reduced matrix indicate that the vectors are linearly independent? I know that a set of vectors is linearly independent when a linear combination of them equal to zero is only satisfied by coefficients that are all zero.
For this particular question, we have a matrix:
$$ \left[
    \begin{array}{ccc|c}
      2&1&7&0\\
      1&-2&-4&0\\
      -2&7&17&0\\
    \end{array}
\right] $$
which I reduced to:
$$ \left[
    \begin{array}{ccc|c}
      1&0&2&0\\
      0&1&3&0\\
      0&0&0&0\\
    \end{array}
\right] $$
My question:
How does this reduced matrix indicate whether or not my vectors are linearly independent, i.e., how do I interpret a matrix like this?
 A: Since the columns of a matrix $A$ span its range, it follows that they are linearly independent if and only if the range of $A$ has maximal dimension, which happens if and only if $A$ has maximal rank.
A similar reasoning applies to the rows of $A$, which span the range of $A^T$.

In particular, a matrix cannot have maximal rank if one of its columns or rows contains only zeros (because in this case the dimension of the kernel is greater than $1$).
A: If 
$$
v_1 = \pmatrix{1 \\ 0 \\ 0}, v_2 = \pmatrix{0\\1\\0}, v_3 = \pmatrix{2\\3\\0}
$$
Then $v_3 = 2v_1 + 3v_2$.
So $v_3$ is a linear combination of $v_1$ and $v_2$ and thus are the vectors not linearly independent.
A: A solution $x$ to $A x = 0$ satisfies 
$x_1 a_{.1} + \cdots + x_n a_{.n} = 0$, 
where $a_{.i}$ is the $i$-th column vector of $A$.
If those column vectors are linear independent then only zero $x_i$ should be possible. This means $Ax=0$ has only the solution $x=0$.
This is the case if your reduced augmented matrix is of the form $[I_n \mid 0]$, where $I_n$ is the identity matrix for $n\times n$ matrices.
Here your augmented matrix is not of that form, so there are infinite solutions of $Ax=0$ and the column vectors are linear dependent.
