A vector by definition is an element of some vector space. Unless specified otherwise, this is the definition you should have in mind. Now let me try to clear up some of your specific questions.
- Here by VECTOR we do not mean the vector quantity which we have defined in vector algebra as a directed line segment.
Without context, it's impossible for me to figure out exactly what is being meant here. Most likely, they previously introduced a specific vector space, such as $\mathbf{R}^n$ and now they want to discuss a different vector space where direction may not have a clear definition.
- Matrices having a single row or column are referred to as vectors.
This is a bit more advanced than what you are probably studying. It basically comes down to how matrices actually arise. Once you fix a basis for your vector space, there is a bijective correspondence between linear transformations and matrices. Then all matrices arise as such. The proof involves taking a basis for the domain and then the columns (or rows) are the images under this map. Well, the image is an element of the codomain, i.e. an element of a vector space, so we can call it a vector.
This way, we can see that all columns of such a matrix is a vector of the codomain. For rows, now just switch the codomain and the domain.
- I also watched a video in which at approximately 3:55 he says that a point in two dimensional real coordinate space is written in matrix form in LINEAR ALGEBRA.
This is going back to 1, where we are once again working in what appears to be $\mathbf{R}^2$. He says it is more common to write the vector $(5,0) \in \mathbf{R}^2$ as a column matrix instead of as a point notation. This is merely a naming or a left vs. right ($xA = b$ vs. $Ax = b$ if you will) and has nothing to do with whether it's a vector.
Cyclic, or circulant matrices are matrices of the following form. So the rows and colums are obtained by cyclical permutation. Since the zero matrix and the identity matrix are cyclic, such matrices may or may not be invertible.
Link: Wikipedia.
Best Answer
$\Vert x\Vert$ is the norm of $x$. Intuitively it's the "length of the vector", or the "distance of the point from the origin".
In your case, you're implicitly working with $\Vert x\Vert = \sqrt{x_1^2+x_2^2+\cdots+ x_n^2}$. You'll see that this is pythagoras' theorem to get the length of a diagonal in a right angled triangle in $n$-dimensions. There are other ways you can define $\Vert x\Vert$, but the one defined above enables you to have a notion of angles in $\mathbb{R}^n$, so it's the standard unless otherwise stated.
If you think of it as the distance of the point from the origin, it's clear that "$\Vert x \Vert ≤ r$ for all $x\in T$" is a sensible definition for boundedness, because it's saying "no points in $T$ are more than $r$ away from the origin".