Vectors

Vectors are a way of describing variables which contain multiple dimensions. It is commonly said that they have both direction and magnitude, while scalar values have only magnitude.

They can be used in many scenarios, but the most common is probably to describe a point, or a transformation in an $N$ dimensional space.

A key point is that vectors are effectively matrices in the context of linear algebra. By which I mean they are presented in the same way, and can be combined (and often are) in the same calculations one might find matrices.

The main difference between a vector and a matrix, is that where matrices can have any number of columns or rows, a vector will have either one row, or one column.

The shape of a matrix is often given by:

$$\textrm{rows}\times \textrm{columns}$$ (3)

... so while a matrix might be $M\times N$, a vector will either be $1\times N$, or $N\times 1$.

If a system makes use of vectors which are presented as a row ($1\times N$), they are called row major systems. It follows then that a system which presents vectors as columns ($N\times 1$), are called column major systems.