Linear Independence and Subspaces 
I'm currently looking at problem #2. I wrote $W$ in the form
$$W=a\begin{bmatrix}
2\\ 1\\ 4\end{bmatrix}+b\begin{bmatrix}1\\ 0\\ 1\end{bmatrix}+c\begin{bmatrix}-1\\ 1\\ 
 1\end{bmatrix}$$
and then I took the determinant of the matrix formed by these vectors, that is,
$$\det\begin{bmatrix}2 & 1 &-1 \\  1&  0& 1\\  4&  1&1 \end{bmatrix}=0$$
Yet the study guide solutions say
$$ W=Col \begin{bmatrix}2 & 1 &-1 \\  1&  0& 1\\  4&  1&1 \end{bmatrix}$$
is a subspace of $\mathbb{R}^3$. I don't understand how it can be if these vectors are not linearly independent as the determinant is zero. What am I missing?

The fact that they are not linearly independent does not mean that W is not a subspace. It is only mean that they are not the basis of the subspace - they aren't the minimal set that span w. . In order to show that W is subspace, you only need to take two vector, u and v in w, and show that u+v contains in w.
