How do you calculate the dimensions of a vector space more generally.
For any field $K$ and $n \in \mathbb{N}$, $M_{n}(K)$ is an algebra over K. The notes says that the vector space dimension is $n^2$.
This is the part of linear algebra I can't get. Please help. Extremely stuck.
I generally know the answer, but don't see the motivation behind the word dimension.
To me a dimension is something like allowing a Euclidean vector some sort of movement i.e. $v=(t,0,0,0)^T$, however in $M_n(K)$ you have no way of getting a vector like that. So what does it mean?