A Guide to Linear Algebra for Data Science

Linear algebra is defined as the branch of mathematics dealing with linear combinations using arithmetic on columns of numbers called vectors and arrays of numbers called matrices. Comprehending linear algebra is necessary for a thorough understanding of machine learning. Linear algebra is Robin to Data Science's Batman. It drives important branches of data science, like the booming areas of computer vision and natural language processing.

Most artificial intelligence models, as well as datasets, are frequently shown as matrixes. Data preprocessing, data transformation, and model validation all involve linear algebra. All the robust machine learning algorithms that we are so accustomed to are based on linear algebra. It is an essential component of a data scientist's toolkit. Regarding Data Science Bootcamp Training and its reviews, one can follow data science bootcamp review.  

One of the key areas of mathematics is linear algebra. The primary focus of linear algebra is the studying of vectors and linear functions. In the way that geometry is presented nowadays, linear algebra is seen as a fundamental idea. To enquire about information related to courses offered, check out the best data science courses.

It is mostly utilized throughout physics and engineering since it helps define fundamental objects like planes, lines, and object rotations. It has computational efficiency and enables us to model a wide range of natural events. 

Essentially, it is seen that the analysis of linear combinations is known as linear algebra. To carry out the linear transformations, knowledge of vector spaces, lines, and planes, as well as some mappings, is necessary. It contains linear functions, matrices, and vectors. It is the investigation of the transformational properties of linear sets of equations.

Why learn Linear Algebra?

Linear algebra is very important when it comes to data science. From the notations being used to define the algorithms to their actual implementation, linear algebra is the basis of machine learning. The significance of linear algebra for machine Data Science is discussed here. 

Below is the image of a cute dog that can be easily perceived by human eyes, but in the case of computers, recognition is not as easy.

Matrix-based tools are used to accomplish this result. Pixels are tiny, indivisible building blocks that make up a digital image. An intensity value is used to describe each pixel in an image. As a result, an image is effectively a matrix with each pixel's intensity values as its constituents. 

‍Most likely, linear algebra is used to expand, compress, crop, or carry out any other action on these photos. The foundation of deep learning and machine learning is linear algebra. Knowledge of these ideas at the vector and matrix levels broadens your viewpoint on a specific ML topic and enriches your understanding of it. Know more about data science learning path. 

Representation of problems in Linear Algebra

Let's begin with a straightforward question. Consider that the cost of one ball and two bats or one bat and two balls is 100 units. The cost of a ball and a bat must be determined. 

Assume that a bat costs Rs "a" and a ball costs Rs. "b." Since "a" and "b" are variables, their values can change depending on the circumstances. 

Let's write the mathematical form  

2a+b=100..........(1)        

Same for the second condition  

a+2b=100..........(2) 

We now require the values of "a" and "b" such that it fulfills both equations in order to determine the costs of the bat and ball. Finding these values of "a" and "b," or the answer to a series of linear equations is the fundamental challenge of linear algebra. 

Visualization of the problem

Flat objects are represented using linear equations. We'll start with the easiest to comprehend, which is line. The collection of all the points that fulfil the given equation is the equation's line. For instance, 

Our equation (1) is satisfied by the points (50, 0), (0, 100), (100/3,100/3) and (30, 40). Therefore, these points ought to be on the line that follows our equation (1). Like that, some of the places that fulfill equation are (0,50), (100,0), (100/3), and (100/3) (2). 

In this instance, we want the point that lies on both lines to satisfy both conditions. The intersection of the two lines, as indicated in the diagram below, is where we naturally want to look. 

2a + b = 100 .......... (1) 

a + 2b = 100 .......... (2) 

By using simple algebraic procedures like addition, subtraction, and substitution, the Third Equation can be calculated as follows 

a + 2*(100-a)/2 = 100......(3) 

Visualization of the Planes

In general, a plane is represented by a linear equation with three variables. Technically speaking, a plane is a flat, infinitely long geometric object. Finding solutions to a linear equation with three variables entails locating the point where those planes join. 

 A set of three planes can intersect in one of four ways: 

• Absolutely no intersection 
• Planes intersecting in a line. 
• They might intersect in a plane. 
• At one point, all three planes intersecting. 

Thus, for solving and visualizing higher dimensional data we need matrices.

Solving The Problem

Writing related objects together in a matrix allows us to quickly manage and modify them as needed. When training different algorithms in data science, it is typically utilized to store input like weights in an artificial neural network. 

Typically, the letters "I" and "j" stand for rows and columns, respectively. The 'i'th row and 'j'th column are used to index the elements. We use a certain alphabet, such as A, to denote the matrix and its components (ij). 

For example 

A[2,1]=4 

Matrix Equations

As indicated below, Lets calculate the result of the matrix multiplication. 

The expression's inclusion of our three equations can be simply confirmed. Our matrices are being called as "A," "X," and "Z”. It clearly shows that we can put all our equations in one place and write them together as 

AX = Z 

We will now examine the two approaches to solving matrix equations in more depth. 

• Inverse of a Matrix 
• Row Echelon Form 

Inverse of the Matrix

An invertible matrix is one for which the inverse matrix can be determined and whose determinant is non-zero 

The multiplicative identity is obtained by multiplying the provided matrix by the other matrix that serves as the inverse of the matrix. A matrix's inverse, A-1, is equal to the matrix's identity, I, therefore A.A-1 = A-1A = I for matrix A.  

For example 

In the case of real numbers, the number a-1 served as the inverse of any real number a, such that a multiplied by a-1 = 1. As long as the number wasn't zero, we knew that the inverse of a real number was equal to the reciprocal of the number. The matrix that results from multiplying A by its inverse, A-1, is known as the identity matrix. 

Matrix A's inverse matrix formula is as follows: 

A is a square matrix, and A-1 = adj(A)/|A|; |A| 0. 

Note: For a matrix's inverse to exist: 

• The provided matrix must be square. 
• It is not acceptable for the matrix's determinant to equal zero. 

Below image depicts the process for finding inverse of 2x2 matrix 

Row Echelon Form

• If a matrix possesses the qualities listed below, it is in Row Echelon form. 
• The matrix's bottom row will always be made up completely of zeros. 

The first non-zero element for each row that would not consist entirely of zeros is 1. (Defined as leading 1). When two (non-zero) rows follow one another, the leading one in the higher row is located more to the left than the leading one in the lower row. 

In reduced row echelon form, every row's first row has 0 in the column below and above it. 

Below is a depiction of row-echelon form:  

And it’s reduced row-echelon form: 

Gaussian elimination, a method, can be used to convert any matrix into reduced row echelon form. For the solution of systems of linear equations, this is especially helpful. 

A matrix can be transformed into the reduced row echelon form using the Gaussian Elimination method. Additionally, it can be employed to track down an answer to the system of linear equations. This is done with the intention of doing various arithmetic computations on the row until there is just one variable left. 

The following are some of the operations we can carry out: 

• Switch any two rows. 
• Together, add two rows. 
• Add a non-zero constant to one row, such as 1/3, -1/5, or 2. 

Given the following linear equation: 

And the augmented matrix above  

This must now be transformed into the row-echelon form. We must use Gaussian Elimination to transform this into row-echelon form. 

To get 0 in the first place of r2 and r3, we must first subtract 2*r1 from r2 and 4*r1 from r3. 

To get the second 0 in the third row, we will first swap the rows r2 and r3, then subtract 5*r2 from r3. 

The value of z may now be calculated from r3, i.e., 10 z = 0 z=0. We can set it to r2, y = 2, using the result of z = 0. Similar to this, by entering the values of y and z in r1, we obtain the value of x=3. 

Rank of the matrix

The number of non-zero rows in the row echelon form determines the matrix's rank. We must take the following actions in order to determine the rank 

• Find the given matrix's row-echelon form. 
• The number of non-zero rows is counted. 

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Application of Linear Algebra in Data Science

Following are the applications of linear algebra for data science fields 

Machine Learning

Loss functions, regularization, support vector classification, and many other machine learning techniques are all uses of linear algebra. 

Loss Function

Linear algebra is employed in loss functions, a particular technique in linear algebra. Your prediction models' accuracy can be assessed using the loss function. 

The loss function will produce a greater value if your model is totally inaccurate. Whereas the loss function would produce a lower value if it were a good one. 

Computer Vision

Linear algebra is utilized in applications like image recognition, some image processing methods like image convolution, and image representation as tensors, (or as we refer to them in linear algebra, vectors)  

Dimensionality Reduction

Singular-Value Decomposition is the most well-known method for reducing the dimensions of data (SVD). A matrix can be broken down into its component pieces using the SVD matrix decomposition technique, which simplifies matrix calculations. 

Natural Language Processing

The branch of artificial intelligence known as "natural language processing" (NLP) is concerned with how computers and people interact while utilizing natural language.

To quantitatively represent the test data, we employ Linear Algebra in NLP algorithms on text. Techniques used are word embeddings that preserve the context of the words in the document by representing them as vectors of numbers. 

Reasons Data Scientists Need Linear Algebra

Reasons why Data Scientists need linear algebra are as follows: 

Machine learning

Loss functions, regularization, support vector classification, and many other machine learning techniques are all applications of linear algebra. 

Computer Vision

Linear algebra is utilized in applications like image recognition, some image processing methods like image convolution 

Natural Language Processing

Natural language processing (NLP) is entirely based on math. Numerous NLP algorithms, including those for text summarization, page ranking, and information retrieval, are based on linear algebra. 

Conclusion

So, in this way, we saw the importance of linear algebra for data science and other applications. We also studied its various methods of representation of problems and various reasons for its importance for the data science/data analysis profession. If one wants to inculcate Data science tools in their skill set, check out the Knowledgehut data science bootcamp review and take an informed decision.