# 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→𝔽 (where 𝔽 is the underlying field)." I do not understand how linear functions V→𝔽 can be called as vectors. I can go to the extent of reading references but i failed a few times and need some advice.

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.

• @joriki Just out of curiosity what happens when you think of the concept vector? Does an example come to mind, the definition or something different? – Mark Jul 13 '11 at 8:33
• @Mark: an arrow :-) – joriki Jul 13 '11 at 8:50
• @joriki I was hoping for something concrete like an example, but I guess an arrow is as concrete as it gets. :) – Mark Jul 13 '11 at 9:10
• @Mark: I think the things that come to my mind when I think "vector" are roughly an arrow, a column vector of reals, and the definition of a vector space, in that order. I should add that my academic training was a theoretical physicist :-) – joriki Jul 13 '11 at 9:40
• @Mark A concrete way to think about a vector applied at a point is to imagine the a kinetic force, for example if someone punched you in the face. The point of application would be the point of contact and the infinitesimal vector (a large norm would correspond to a hard hit) represents the directed energy osculating your face. Abstract vectors allow for many more examples than just forces. – isomorphismes Aug 7 '13 at 4:55

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.

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.