What exactly does linear dependence and linear independence imply? I have a very hard time remembering which is which between linear independence and linear dependence... that is, if I am asked to specify whether a set of vectors are linearly dependent or independent, I'd be able to find out if $\vec{x}=\vec{0}$ is the only solution to $A\vec{x}=\vec{0}$, but I would be stuck guessing whether that means that the vectors are linearly dependent or independent.
Is there a way that I can understand what the consequence of this trait is so that I can confidently answer such a question on a test?

I've found the best way to understand this is as follows: a set of vectors is linearly dependent if you can write one of them in terms of the others.
When you multiply a matrix by a vector, $A\vec{x}$, that's shorthand for "multiply each column of $A$ by the corresponding entry of $\vec{x}$, and then add them together." If the columns in $A$ are linearly dependent, then there's some $\vec{x}$ that will allow them to cancel, giving you the zero vector.
