Overview of Linear Algebra Very new student tackling this course, and I've never been this terrified from Math before. I cannot grasp the meaning of things in Linear Algebra, most of what's stated is either obscure, meaningless, and abstract when I am first tackling them. What sort of advice can you give?
I'll be more specific and state my misconceptions.
First and foremost, the simplest question of all; What in the world is a subspace? I  am NOT looking for the criteria that makes a subspace so. I want to cling into any fundamental definition. Why for instance should it include the 0 vector? What does a subspace even look like (assuming we are working with R2 and R3) 
When I am told that it possess closure under addition and multiplication, I always imagine an infinite plane or a line -- basically; the coordinate space as whole (If that really is the case, then what is the point of defining a subspace). 
These lingering thoughts are really dragging me behind...I have a LOT more questions to come, but let this be an impression.
Thanks for Clarifying~

First of all, while it is good to have an intuition and visualization for what mathematical objects are, you cannot remove yourself from the formal definitions. When things get too complicated to visualize, knowing and being able to apply definitions will save you.
That being said, recall that a vector space $X$ is a set which satisfies a long list of axioms. A subspace of $X$ is a subset of $X$ which also satisfies all those axioms. One of the axioms is that a vector space must include a zero vector. Thus, any subspace must also include the zero vector.
A subspace of $\mathbb R^2$ is either just the origin or a line through the origin (or even all of $\mathbb R^2$ is a subspace of itself). A subspace of $\mathbb R^3$ is either the origin, a line containing the origin, or a plane containing the origin (or again, all of $\mathbb R^3$). The reason for defining a subspace is that not all vector spaces can be visualized at $\mathbb R^2$ or $\mathbb R^3$. Some are vastly more complicated. By having a formal definition, we can find properties true of all subspaces, not just planes and lines.
