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Dot Product
Prerequisites
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Matrices
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Definition
A matrix is a rectangular group of numbers (or symbols) with a given number of rows and columns.
Example

The following matrix has two rows and three columns, and is described as a "two by three" (2x3) matrix.

LaTeX
\begin{bmatrix}
    3 & 6 & 2  \\
    5 & 1 & 10
\end{bmatrix}
Notation
  • There are a few different ways to represent a matrix. The easiest is with an bold uppercase letter: A
Code
A = [[3, 6, 2], [5, 1, 10]]

print(A)
Notes
  • The plural of matrix is matrices.
  • The size of a matrix is called its dimension or order. It is written with as rows x columns. A matrix with 2 rows and 3 columns is written as 2x3.
  • Two matrices are considered equal if and only if:
    • They have the same number of rows.
    • They have the same number of columns.
    • Each corresponding element is equal. i.e. x111 = x211
  • The numpy.matrix is depreciated, in favour of regular arrays
Matrix Transposition
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Introduction
  • Matrix Transposition is an operation that converts one matrix into another.
  • Each row in the original matrix is used as a column in the transposed matrix.
Notation
There are two main notations used to show that a matrix is transposed:
  • Using a prime: the transpose of matrix A is A'
  • Using a T superscript: the tranpose of matrix A is AT
Example
Code
import numpy as np

a = np.matrix([[3, 6, 2], [5, 1, 10]])
print(a)

transposed = a.transpose()
print(transposed)
Notes
  • The order (or dimension) of a matrix is switched when transposing, i.e. a 3x2 matrix creates a 2x3 matrix.
Vectors
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Introduction
  • Vectors are special types of matrices which have only one column or row.
Example
Notation
  • A column vector is represented by a bold lowercase letter, e.g. a
  • A row vector is a transposed column vector. It is represented by a bold lowercase letter with a prime, e.g. a'
Code
In python, both row and column vectors are represented as arrays.
a = [3, 6, 2]
print(a)

Dot Product

Introduction
  • The dot product is a method of multiplying two vectors and receiving a number as the result.
  • The first vector must be a row vector, while the second must be a column vector.
  • Both vectors must contain the same amount of elements.
  • To calculate the dot product, multiply the corresponding elements and sum the results.
Mathematical Definition
a and b are column vectors.
Example
Code
import numpy as np

a = [3, 6, 2]
b = [1, 3, 8]

c = np.dot(a, b)
print(c)
Notes
  • The dot product is also called the 'vector inner product' or the 'scalar product'.
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