Vectors

What is it?

Vectors are one of the most used objects in Mathematics, being applied in Tensors, Linear Algebra, Machine Learning, and Deep Learning.

Vectors refers to quantities that cannot be expressed by a single value, but its specific meaning can change from where it’s been applied. For example, in Euclidean Geometry, a vector is an object with magnitude and direction, that goes from point $A$ to point $B$, represented by a line and denoted by $\overline{AB}$.

In Linear Algebra, vectors can be thought as single row or single column Matrices. A row vector contains a single row and a column vector contains a single column. For example:

$$\begin{array}{rl}& \text{This a row vector: }\left[\begin{array}{ccc}-1& 2.52& x^2\end{array}\right]\\ & \\ & \text{This is a column vector: }\left[\begin{array}{c}0.75\\ 50\\ \sqrt{y}\end{array}\right]\end{array}$$

There are two common notations for vectors. A bold lowercase letter, or a lowercase letter with an arrow over it. For example:

$$\begin{array}{rl}& \text{Bold lowercase notation: }\text{x , y, z}\\ & \text{Overline lowercase notation: }\overline{x},\overline{y},\overline{z}\end{array}$$