Matrix and vectors, why different notation for dimensions? If we collect data and put it into a matrix of size (100,3), we tend to say we have three-dimensional data. We think of each column as a dimension.
On the other side, if we have a vector of size (100,1), we tend to say this is a hundred-dimensional vector.
But the vector is also a matrix. In the case above, we can say this is a (100,1) matrix or (100,1) vector. Yet in the matrix case, we would say it is one dimensional, but for vector, we would say it is a hundred dimensional. Why?
 A: I don't agree with the premise that

if we have a vector of size (100,1), we tend to say this is a hundred-dimensional vector.

I would say it is a vector of length 100.
A vector can certainly be considered a matrix, and in linear algebra we can think of a vector as a column vector, in for example:
$$ y = Xb$$
where $X$ is a matrix of dimension $n \times m$ and $b$ is a column vector of $m \times 1$ dimension, resulting in a $n \times 1 $ column vector. This is the familiar scenario in regression where $X$ corresponds to the data and we would naturally think of there being $n$ observations, and $m$ variables / features / dimensions. We can think of each column of $X$ as a column vector, of length $n$, representing one of the variables in the data.
Alternatively, a vector can be a row vector. For example, the dot product of two vectors, $ x \cdot y $, where $x$ is a $1 \times n $ row vector and $y$ is a $n \times 1 $ column vector.
A: If your data is $n \times m$ matrix, where $n$ is the number of samples and $m$ the number of features, it’s $m$-dimensional data. If $m$ is 3, it’s three-dimensional, if it’s 1, it’s unidimensional. With one column, it’s a column vector, so it has one dimension. Other case may be when someone is describing each sample in terms of a (row) vector of length $m$, so you have $n$ such vectors and data is $m$-dimensional. Dimensionality is about number of features, not samples.
