undetstanding of Vectors, Tangent Vectors, Tangent Covectors To my limited knowledge, I only know vector as a certain fixed number of real numbers put together. for example $[1,2.3,6.4,0.75]$ is a vector. A vector of dimension $N$ is any of the elements of the set $\underbrace{\mathbb{R} \times \mathbb{R}\times.....\mathbb{R}}_{N \text{ times}}$. Ok, I also know the that sequences and functions of a real variable can also be vectors, for example the collection of all square integrable functions or the collection of all square summable sequences can be vector spaces. I can imagine the notion of addition of vectors in all these examples as addition of corresponding elements of two vectors to form the corresponding element of the sum vector, for example the addition of two square integrable functions $f$ and $g$ is nothing but the pointwise addition of values of the function to form the values of the resultant sum function $f+g$.
The point-wise addition is a must for me to imagine a vector. But I am finding hard and clueless when I try to read and understand concepts like tangent vectors and tangent co-vectors. I am clueless and I can't even try to explain my difficulty, hope someone understands my problem and put things for me so that i can overcome this. I was also reading this answer by Aaron here but i am very far from understanding things like "These tangent vectors act on functions by taking the directional derivative of a function at a point. If you take a tangent covector, it no longer acts on functions, it just acts on vectors. " and "a "dual" space V∗ which consists of linear functions V→ (where  is the underlying field)." I do not understand how linear functions V→ can be called as vectors. I can go to the extent of reading references but i failed a few times and need some advice.
 A: The main thing to understand here is that "vector" merely refers to any element of any vector space. Anything that can be added together and multiplied by a scalar (with the appropriate conditions on these operations) is a vector; there's no requirement of points or components. Linear functions are vectors because you can add them together and multiply them by scalars. That's all there is to it. Trying to "imagine a vector" merely distracts you from what's going on.
A: Tangent vectors are applied at a particular point. Let's say you are trying to hit a tennis ball, then you need to apply your 3-vector of force to where the ball actually is. Connecting with the ball would attach your force-vector to the contact point of the surface of the ball.
tennis http://www.mathematik.uni-kl.de/fileadmin/agag/wirthm/Bilder/Top2.jpg
Also logically a tangent space is where derivatives on a manifold can "live" -- i.e., where they can be added and subtracted while being "of the same type". (We don't add apples and oranges, and we don't add kicks to the groin to punches in the face either.)

Covectors are weighted sums. Imagine you played around with the weighting scheme $a,b,c,d$ in $$a \cdot 1 + b \cdot 2.3 + c \cdot 6.4 + d \cdot 0.75 \quad = \quad ?$$. That would lead to different sums but you'd be changing "parameters rather than inputs". Of course there's not a huge difference which is why the concepts are so easily confused.
