What values of a is the set of vectors linearly dependent? The question is is "determine conditions on the scalars so that the set of vectors is linearly dependent".
$$ v_1 = \begin{bmatrix} 1 \\  2 \\  1\\  \end{bmatrix}, v_2 = \begin{bmatrix} 1 \\ a \\ 3 \\ \end{bmatrix}, v_3 = \begin{bmatrix} 0 \\ 2 \\ b \\ \end{bmatrix}
$$
When I reduce the matrix I get
$$\begin{bmatrix} 1 & 0 & 0 \\ 0 & a-2 & 0 \\ 0 & 0 & b - \frac{4}{(a-2)} \end{bmatrix}$$
If the matrix is linearly independent then shouldn't $a-2 = 0$ and $b - \frac{4}{(a-2)} = 0$? So, I said the solution is when $a-2 \neq 0 $ and $b - \frac{4}{(a-2)} \neq 0$. The textbooks says the answer is when $ b(a-2) = 4 $. I understand how they got to $ b(a-2) = 4 $ but why is it equals instead of not equals?

The Determinant Test is appropriate here, since you have three vectors from $\mathbb{R}^{3}$. The set of vectors is linearly dependent if and only if $det(M) = 0$, where $M = [v_{1} v_{2} v_{3}]$. 
