Introduction
Linear algebra is the bedrock of data science and machine learning. Linear algebra provides mathematical insights to solve equations we use to build models.
Actually, sometimes people completed their basic courses on machine learning but when it comes to implementation of the model for huge volumes of data sets than the big problem occurs how to implement particular algorithms and what is the mathematics behind it.
Linear algebra gives the better intuitions about machine learning and data science.
For the better understanding of linear algebra, I would like to present different uses of linear algebra in machine learning and data science.
Let suppose you are working with text data for text analysis and you heard about Bag of words, TF-IDF, and all these counts the value of text present in the text file or document and later it could be used for Language generation, semantic analysis etc.
Another example comes to mind from image processing how a computer stores an image and its attributes and recognize it. it’s all due to the matrix representation of data in 0 and 1 format of pixel intensity.
Other areas are working in high dimensional data sets and want for feature extraction we generally use Principal component analysis and Singular value decomposition these all based on linear algebra. PCA uses the concept of Eigenvalue and Eigenvectors.
What Is Linear Algebra
Linear algebra is about working on the linear system of equations.
Let suppose an example the price of 10 pen and 2 pencil is 24 and 5 pen and 3 pencil is 16 on the basis of these data we want to find the unit price of pen and pencil, So we put these values into a linear equation like
let suppose pen = x and pencil=y
10x+2y=24-----------------------(1) 5x+3y=16-------------------------(2) after solving these two equations we found that the unit price of pen is 2 and pencil is also 2. so when we also try to represent it in the form of matrix form we can write as, [10,2][x]=[24] [5 ,3][y]=[16]
After seeing this we can say that in linear algebra data is in form of linear equations and these linear equations, in turn, represent in form of matrices and vector spaces.
Important terminologies in linear algebra
Scalar: In linear algebra, the scalar is just a single number with have no dimension it has magnitude only.
import numpy as np
num=np.array(25)
print("Number is ",num)
print("Shape of scalar is ",num.shape)
Output:Number is 25
Shape of scalar is ()
Vectors: Vector is a one-dimensional array either in row way or column way. This is also called as an array of numbers, arranged in order and every element identified by an index. In python, list element generally a type of vectors.
num=np.array([1,2,3,4,5,6])
print("Number is ",num)
print("Shape of vector is ",num.shape)
Output:
Number is [1 2 3 4 5 6]
Shape of vector is (6,)
Matrix: Matrix is a two-dimensional array for which we have rows and columns. If our matrix is said to be m x n matrix, it has m rows and n columns. Matrix is the core structure data science, machine learning, and deep learning.
num=np.array([[1,2,4,5,6,7],[7,2,4,6,5,1]])
print("Number is ",num)
print("Shape of vector is ",num.shape)
Output:
Number is [[1 2 4 5 6 7]
[7 2 4 6 5 1]]
Shape of matrix is (2, 6)
Tensors: Tensor is more than two dimensional so we can say that vector and matrix is a subset of a tensor. In tensor row extend along with the x-axis and column extends along with the y-axis.
num=np.array([[[1,2,4,5,6,7],[7,2,4,6,5,1],[7,2,4,6,5,1]]]) print("Number is ",num) print("Shape of tensor is ",num.shape) Output: Number is [[[1 2 4 5 6 7] [7 2 4 6 5 1] [7 2 4 6 5 1]]] Shape of tensor is (1, 3, 6)
Operations on Matrix:
Addition: Addition of matrix is similar to normal arithmetic addition, Let suppose a matrix Aij and another matrix Bij and the resultant matrix Cij=Aij+Bij. where i =rows j=columns
Aij=[1,2,3] Bij=[4,6,8] [2,3,5] [5,6,8] then cij=[5,8,11] [7,9,13] Programming Implementation a=np.array([[1,2,3],[2,3,5]]) b=np.array([[4,6,8],[5,6,8]]) c=a+b print(c) Output:[[ 5 8 11] [ 7 9 13]]
Subtraction: Subtraction of matrix is similar to normal arithmetic subtraction, Let suppose a matrix Aij and another matrix Bij and the resultant matrix Cij=Aij-Bij. where i =rows j=columns.
Aij=[1,2,3] Bij=[4,6,8] [2,3,5] [5,6,8] then cij=[-3,-4,-5] [-3,-3,-3] Programming Implementation a=np.array([[1,2,3],[2,3,5]]) b=np.array([[4,6,8],[5,6,8]]) c=a-b print(c) Output:[[ -3 -4 -5] [ -3 -3 -3]]
Multiplication: Multiplication of matrix is different from normal arithmetic multiplications.
Before proceeding the multiplication of the matrix we have to keep some conditions in mind as given below:
- Multiplication of matrix of order m x n and n x p would be the order of m x p.
- Multiplication of matrix would be possible when the number of rows of the first column will equal to the number of columns of the second matrix ie, Aij=Ajk
a=np.array([[1,2],[3,4],[5,6]]) b=np.array([[4,6,8],[5,6,8]]) c=a.dot(b) print("multiplication of matrices ",c) Output:multiplication of matrices [[14 18 24] [32 42 56] [50 66 88]]
Properties of matrix multiplication:
- Associativity : Yes –> AB(C)=A(BC)
- Commutative : No –> AB≠BC
Types of Matrix
Identity Matrix: Type of square matrix having all diagonal elements 1 and nondiagonal 0. Represented as I.
Scalar Matrix: Type of square matrix having all diagonal elements equal to some constants.
Diagonal Matrix: Matrix having only diagonal elements but the diagonal matrix is 0.
Column Matrix: Matrix having only column elements.
Row Matrix: Matrix having only rows elements.
Upper triangular matrix: A square matrix which has lower elements from diagonal are zero.
Lower triangular matrix: A square matrix which has upper elements from diagonal are zero.
Transpose of a matrix
Matrix obtains from any given matrix A, by interchanging rows and columns called the transpose of a matrix A, and denoted by A‘.
In python, we can obtain this by Numpy
A=np.arange(10,20,1).reshape(5,2)
print('Before Transpose A is a {0}X{1} matrix'.format(2,5))
print(A)
B=np.transpose(A)
print('After Transpose A is a {0}X{1} matrix'.format(5,2))
print(B)
Before Transpose A is a 2X5 matrix [[10 11] [12 13] [14 15] [16 17] [18 19]] After Transpose A is a 5X2 matrix [[10 12 14 16 18] [11 13 15 17 19]]
Foot Note
This article is divided into two parts in this part we learned about basic matrix and it’s operations, In next article we will go through some advanced concept like inverse, eigenvalue eigenvector etc.
If you liked this article be sure to like this article and you have any questions related to this answer I will do my best to answer.
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