linear algebra for deep learning: the fundamentals you need to know

Linear Algebra for Deep Learning: The Fundamentals you need to know

12 mins read
Credits: Applied AI course ( YT Channel )

Linear algebra is an important part of deep learning. It’s used in everything from the singular value decomposition to building neural networks, but it can be overwhelming, even if you’re just getting started with machine learning and artificial intelligence.

In this guide, you’ll learn the fundamentals of linear algebra you need to know as a beginner to machine learning and deep learning! By the end of this guide, you’ll know about vectors, matrices, matrix multiplication, and much more!

Basic definitions

In Mathematics, a vector is an ordered list of numbers. In deep learning, we use vectors to represent data points. A matrix is a two-dimensional array of numbers. In deep learning, we use matrices to represent the weights of neural networks.

Deep Learning is a subdomain of machine learning concerned with the artificial neural network, an algorithm that mimics the function and structure of the brain.

A scalar is a single number. In deep learning, we use scalars to represent the bias of neural networks. A tensor is a multi-dimensional array of numbers. In deep learning, we use tensors to represent the data in our datasets.


Vectors – A vector is a numerical array that can be in a row or a column. A Vector has a single index that can point to any value within the Vector. In the graphic above, V2 refers to the second value within the Vector, which is -8.


Matrix – A matrix is an ordered two-dimensional array of numbers with two indices. The first one indicates the row, while the second indicates the column. In the yellow graphic above, M23 refers to the value in the second row and third column, which is 8.

A matrix can have an arbitrary number of rows and columns. It should be noted that a Vector is also a Matrix, but with only one row or column.

Operations with vectors

When working with vectors, you’ll need to be able to perform various operations. These include addition, subtraction, multiplication (scalar and vector), and division. You should also be familiar with the dot product and cross product.

These operations will come in handy when working with deep learning algorithms. For example, the vectorized form of gradient descent is 

which can then be broken down into two steps: updating a weight based on its corresponding gradient by multiplying it by that gradient’s magnitude, and updating all weights after adding this new weight update to them.

Solving systems of equations using matrices

When you’re working with deep learning, you’ll often need to solve systems of equations. And one of the best ways to do that is using matrices.

Matrices are a way of representing data in a tabular form, and they can be used to represent linear equations.

To solve a system of linear equations using matrices, you first need to convert the equations into matrix form.

Then, you can use matrix operations to solve the equations. Matrix operations include addition, subtraction, multiplication, and division. You can also use matrices to find the inverse of a matrix, which is useful for solving systems of linear equations.

Linear independence and bases

Linear independence is a key concept in linear algebra that you need to understand for deep learning. Basically, a set of vectors is linearly independent if no vector in the set can be expressed as a linear combination of the other vectors.

A basis is a set of linearly independent vectors that span a vector space. In other words, a basis is a set of vectors that you can use to represent any vector in the space.

A typical example of this is when you are doing projections or coordinate transformations and want to convert from one coordinate system to another.

When transforming from Cartesian coordinates (x, y) to polar coordinates (r, θ), the basis {(1, 0), (0, 1)} is used because it spans both spaces and hence is linearly independent. If a set of vectors are not linearly independent they are called dependent.

Applications to machine learning


Linear algebra is critical for understanding deep learning. It helps us understand how to represent data, how to transform data, and how to reason about relationships between variables. Additionally, linear algebra provides the foundation for important machine learning concepts such as gradient descent and backpropagation.

In this post, we’ll cover the basics of linear algebra that you need to know in order to understand deep learning.

Learn Linear Algebra Notation You must be able to read and write
vector and matrix notation. The algorithm is described in vector and matrix notation in books, articles, and websites.

Learn Linear Algebra Arithmetic Arithmetic operations are performed in conjunction with
linear algebra notation.

I need to know how to add, subtract and multiply scalars, vectors and matrices.
A challenge for beginners in the field of linear algebra are operations such as matrix multiplication and tensor multiplication.

They are not implemented as direct multiplication of the elements of these structures, which seems counterintuitive at first glance.

Video Credits: TensorFlow

Learn Linear Algebra for Statistics

To learn statistics, you need to learn linear algebra. Especially multivariate statistics.
Statistics deals with the description and understanding of data. As a mathematics of data, linear algebra has left its mark in many relevant areas of mathematics, including statistics.
To read and interpret statistics, you must learn linear algebra notation and arithmetic.

Learn Matrix Factorization


The idea of ​​matrix factorization, also known as matrix factorization, is based on notation and arithmetic.
You need to know how to factor a matrix and what it means.

( Matrix Factorization )Matrix factorization is an important tool for linear algebra and is commonly used as an element of more complex operations in both linear algebra (inverse matrix, etc.) and machine learning (minimum square, PCA, SVD, etc.). ..

Learning Linear Least Squares

Credits: StatQuest with Josh Starmer ( YT Channel )

I need to know how to solve linear least squares using matrix factorization.
Linear Algebra was originally developed for solving systems of linear equations. These are equations with more equations than unknown variables.

As a result, they are arithmetically difficult to solve because there is no single solution, since there is no line or plane that can perfectly fit the data.
This type of problem is called least squares minimization and can be restated in the language of linear algebra called linear least squares.

Deep learning – Dozens of passes through neural networks

We can see linear algebra in action in all major applications today. Examples include sentiment analysis on a LinkedIn or Twitter post (embedded), detecting a type of lung infection from X-ray images (computer vision), or any text-to-speech bot (NLP). ).

All of these data types are represented by numbers in tensors. We perform vectorization operations to learn patterns from them using neural networks. It then generates a processed tensor, which in turn is decoded to generate the final model inference.

Data Representation:-

Data Representation Data, the fuel for ML models, must be converted into arrays before it can be fed into your models. These arrays are used for computations such as matrix multiplication (dot product). This returns the output, which is also represented as a transformed matrix/tensor of numbers.

An industry where linear algebra is heavily used:-

We hope you are confident that linear algebra is driving ML initiatives in various areas today. If not, here is a list with some examples:
1. Chemical physics
2. Genomics
3. Word embedding – neural network / deep learning
4. Robotics
5. Image processing
6. Quantum physics


In the world of deep learning, linear algebra is the mathematics of computation. It is the study of vectors and matrices and their properties. Linear algebra is a critical tool for understanding deep learning algorithms and how they work.

Without a strong foundation in linear algebra, it will be difficult to understand and implement deep learning algorithms. In this blog post, we will cover the basics of linear algebra that you need to know in order to get started with deep learning.

We will cover topics such as vector operations, matrix operations, eigenvalues and eigenvectors, singular value decomposition, and more. By the end of this post, you will have a strong foundation in linear algebra and be well on your way to understanding deep learning algorithms.

FAQ ( Frequently Asked Questions)

Do you need linear algebra for deep learning?

Linear Algebra concepts are critical for understanding the theory behind Machine Learning, particularly Deep Learning. They provide a better understanding of how algorithms work under the hood, allowing you to make better decisions.

What parts of linear algebra are used in machine learning?

Linear Algebra is multidimensional data processing Then, in order to do those things, you’ll end up using pretty much everything else taught in an introductory linear algebra class. But it depends on what you’re doing… different algorithms employ different techniques and math.

Is linear algebra required for machine learning?

You do not need to learn linear algebra to get started with machine learning, but you may want to at some point. In fact, if there was one area of mathematics that I would recommend improving first, it would be linear algebra. 

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