Matrices, matrices and matrices' dimensions

Another good function of matrices is that it shows its dimensions clearly.

Let there be a 3 x 3 matrices and 5 x 5 matrices, M & W:



M has two vectors and each of it has two unit vectors, and every unit vectors has its unique direction. So a vector with two unit vectors is in a two dimensional space. And a vector with five unit vectors, W, is in a five dimensional space.

Now let's see what property a hype-dimensional vector has:

First, what mean by "vertical" here?

We know it very well in a 3-D space, our space, but we know little in a hyper-dimensional space, so what mean by "vertical" here?

Second, what mean by "slope" here?

and lots else coming too. Think about it when you are free=)