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.
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 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 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.
Alternate method: Vector Dot Product
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