Vectors

Chapter: Vector

What is a Vector?

A vector is a fundamental mathematical concept in linear algebra and geometry. It represents a quantity that has both magnitude (length) and direction. Vectors are often used to describe physical quantities such as displacement, velocity, and force.

Example:

Imagine you are navigating through a two-dimensional space, and you want to represent your movement from the origin (0,0) to a point (3,4). You can express this movement as a vector v:

$$v = [3, 4]$$

In this vector, the number 3 represents the horizontal displacement, and the number 4 represents the vertical displacement. The magnitude of the vector, often denoted as ||v||, is calculated using the Pythagorean theorem:

$$||v|| = √(3² + 4²) = √(9 + 16) = √25 = 5$$

So, the vector v has a magnitude of 5 and points in a direction from the origin to the point (3,4). Vectors play a crucial role in various mathematical and physical applications, including deep learning, where they are used to represent features, weights, and data points in high-dimensional spaces.

>>> import numpy as np

>>> x = np.array([1,2,3]) 

>>> print(x)

[123]

>>> print(x.reshape((3,1))) 

[[1] [2] [3]]