Is the set of all differentiable functions $f: \Re \rightarrow \Re$ such that $f'(0) = 0$ is a vector space over $\Re$? I was given the answer yes by someone who is better at math than me and he tried to explain it to me, but I don't understand. I am having difficulty trying to conceptualize this idea of vector spaces with functions because I can't really visualize it like a plane in 3d space. I am also wondering what is the importance of having vector spaces set over a field? It seems trivial or maybe its just me being brainwashed by years of elementary mathematics
[Math] Is the set of all differentiable functions a vector space
linear algebravector-spaces
Related Solutions
This is simply applying a vector space structure to functions from a given set to $\mathbb{C}$. For instance, take $A = \mathbb{R}$. The set of all complex-valued functions on $\mathbb{R}$ form a vector space where scalar multiplication and vector addition are defined as given there. This is simply showing how we can make vector spaces out of functions.
If you have seen the notation $T \in \mathcal{L}(V,W)$, this might be a little easier to grasp. That is read as "The function $T$ is in the set of all linear transformation from $V$ to $W$", where $\mathcal{L}(V,W)$ is the set of all such transformations. This set has a natural vector space structure on it, namely the one given in the snippet you posted:
For $\alpha \in F$ (the underlying field of both $V$ and $W$) define $\alpha T$ as $v \mapsto \alpha T(v)$ and $S+T$ as $v \mapsto S(v) + T(v)$. This gives us a vector space structure, and the objects (vectors) in this vector space are linear transformations (i.e., functions).
Does that help your understanding?
Informally, you can imagine a vector field as a collection of little floating "arrows" attached to points in space. For example, a vector field might represent the velocity of the air in a room: at each point in space, you can ask the question "How fast and in what direction is the wind moving at this point?", and represent that with a vector that is "pinned" (so to speak) to that point in space. The room is then "filled" with little arrows, one at every possible location. The wind velocity at one location is not necessarily the same at another location, so these vectors are not all the same. Nor is it meaningful to "add" vectors that are attached to different points in the room, so the individual vectors don't live in a single vector space.
On the other hand, if you pick a single point in the room, and ask the question "What are all the possible wind velocities at this location?" then you have a vector space. At that one point, there are possible vectors pointing in every direction and with every length. Adding those vectors together is a meaningful operation. At each individual point, there is a vector space associated with that point.
So informally, a vector field can be thought of as choosing, at each point in the underlying space, a single vector from the vector space at that point.
Best Answer
A vector space is merely a set with two operations, addition and scalar multiplication, that satisfy certain conditions. In this case the scalars are real numbers. The addition operation is the pointwise sum, and scalar multiplication is multiplication by a real number.
Besides these properties it geometrically has little to do with vectors in three dimensional space, and the concept of a vector having "both a magnitude and direction" no longer makes much sense (without additional structure). This is the definition of a vector space that has been settled on, so getting used to it would be a good idea.
That said, vector spaces are incredibly well behaved and a great deal of algebraic material from the finite dimensional case generalizes to infinite dimensional vector spaces without modification. This is part of the utility of vector spaces; once you have a set that satisfies a few easy axioms you can do linear algebra in it.