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?
Best Answer
Intuitively vectors being linearly independent means they represent independent directions in your vector spaces, while linearly dependent vectors means they don't. So for example if you have a set of vector $\{x_1, ..., x_5\}$ and you can walk some distance in the $x_1$ direction, then a difference distance in $x_2$, then again in the direction of $x_3$. If in the end you are back where you started then the vectors are linearly dependent (notice that I did not use all the vectors).
This is the intuition behind the notion and you can make it into a definition because in the above example if we start at $0$ then we walk $a_i$ in the $x_i$ direction, then the above paragraph says that $a_1x_1+a_2x_2+a_3x_3=0$. (This is how you should think of linear combinations, as directions to go given by your vectors.)
Finally I will say that you should memorize the definitions. I've taught linear algebra to students that it is their first proof-based math class, and many students don't realize how important knowing the PRECISE definition is. Definitions are crucial and changing one single word can completely change the meaning. So my advice when just starting out is that you should make flash cards of ALL definitions in your book and memorize them. Then once you know them exactly look at the examples after the definition in the book and see how the examples fit the definition.