Vector Basics

Scalars

Scalars are single numbers. Example: x ∈ R denotes that the scalar value x is a member of real valued numbers R.

Vector

Vector in Machine Learning is a collection/array of numbers that corresponds to some features.

Example: [2,5,1] may be used to classify an apple where the first, second and third values represent features such as size, color and number of seeds in a fruit respectively.

Vector in Python can be represented as a NumPy array.

from numpy import array
v = array([2, 5, 1])
print(v)

## Output
[2 5 1]

Vector Arithmetic

Two vectors of equal length can be added, subtracted, divided or multiplied with each other to result in a new vector with the same length.

If a = [a1, a2, a3] and b = [b1, b2, b3] then the following operation will yield:

a) Addition: c = [a1+b1, a2+b2, a3+b3]

b) Subtraction: c = [a1-b1, a2-b2, a3-b3]

c) Division: c = [a1/b1, a2/b2, a3/b3]

d) Multiplication: c = [a1*b1, a2*b2, a3*b3]

# Vector Arithmetic
a = array([10, 20, 30])
b = array([5, 10, 15])

addition = a + b
subtraction = a - b
division = a / b
multiplication = a * b

# print result
print(addition)
print(subtraction)
print(division)
print(multiplication)

[15 30 45]
[ 5 10 15]
[2. 2. 2.]
[ 50 200 450]

Vector Scalar Multiplication

Vector can be multiplied by a scalar value. This results in scaling the magnitude of a vector.

If a = [a1, a2, a3] and s = scalar

Vector Scalar Multiplication: c = [s*a1, s*a2, s*a3]

# Vector Scalar Multiplication
a = array([10, 20, 30])
s = 0.5

vsmulti = s * a
print(vsmulti)

[ 5. 10. 15.]

Vector Dot Product

Vector dot product is a number/value obtained by adding the multiplied elements of two vectors of the same length. Named after the dot(period) operator which describes it.

If a = [a1, a2, a3] and b = [b1, b2, b3] then

Vector Dot Product: c = a . b = (a1*b1 + a2*b2 + a3*b3)

Note: The dot product is an important tool for calculating vector projections, determining orthogonality, etc.

# Vector Scalar Multiplication
a = array([1, 2, 3])
b = array([1, 2, 3])

dotProduct = a.dot(b)
print(dotProduct)

14

Alternate method: Vector Dot Product

import numpy as np
np.vdot(a,b)

14