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Category: Linear Algebra

#010 Linear Algebra – Linear least squares

Highlight: Linear least squares is a very powerful algorithm to find the approximate solutions of overdetermined linear systems of linear equations. Those are systems of linear equations that have more equations than unknowns. The solution to this idea is to minimize the sum of squares of errors in the equation. This method was discovered independently by the mathematicians, Carl Friedrich Gauss, and Adrien-Marie Legendre, around the beginning of the 19th century. So, let’s begin with…
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#011 Linear Algebra – Nonlinear Least Squares

Highlights: In the real-world scenario, models don’t produce linear graphs that often. Most of the time the equation of the model involves higher-order and higher-degree functions. In this post, we will learn how to solve the harder nonlinear equations using a heuristic algorithm of finding the least-squares approximate solution. So let’s begin. Tutorial Overview: Nonlinear Equations Introduction Difficulty Of Solving nonlinear equations Nonlinear Least Squares Optimality condition Gauss-Newton Algorithm Basic Gauss-Newton Algorithm Shortcomings Of The Gauss-Newton Algorithm…
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#007 Linear Algebra – Change of basis

Highlight: So far, we have already talked that it is possible to represent the vector using different basis vectors. In this post we will learn how to go from our standard coordinate system \(\left ( x,y \right ) \) into some other bases. Next, we will also learn why this change of basis can be very useful. For now, we will just say that it’s frequently applied in many signal processing and machine learning methods.…
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#006 Linear Algebra – Inner or Dot Product of two Vectors

Highlight: In this post we will review one of the fundamental operators in Linear Algebra. It is known as a Dot product or an Inner product of two vectors. Most of you are already familiar with this operator, and actually it’s quite easy to explain. And yet, we will give some additional insights as well as some basic info how to use it in Python. Tutorial Overview: Dot product :: Definition and properties Linear functions…
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#008 Linear Algebra – Eigenvectors and Eigenvalues

Highlight: In this post we will talk about eigenvalues and eigenvectors. This concept proved to be quite puzzling to comprehend for many machine learning and linear algebra practitioners. However, with a very good and solid introduction that we provided in our previous posts we will be able to explain eigenvalues and eigenvectors and enable very good visual interpretation and intuition of these topics. We give some Python recipes as well. Tutorial Overview: Intuition about eigenvectors…
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