We will see two important matrices: the identity matrix and the inverse matrix. We will see why they are important in linear algebra and how to use it with Numpy. Finally, we will see an example on how to solve a system of linear equations with the inverse matrix. Linear Dependence and Span. In this chapter we will continue to study systems of linear equations. We will see the intuition, the graphical representation and the proof behind this statement. We will also see what is linear combination. Finally, we will see examples of overdetermined and underdetermined systems of equations.
The norm of a vector is a function that takes a vector in input and outputs a positive value.
Matrix Operations for Engineers and Scientists: An Essential Guide in Linear Algebra
It is for example used to evaluate the distance between the prediction of a model and the actual value. Special Kinds of Matrices and Vectors. We will see other type of vectors and matrices in this chapter. It is not a big chapter but it is important to understand the next ones. We will see some major concepts of linear algebra in this chapter. We will start by getting some ideas on eigenvectors and eigenvalues.
Conditions of Use
We will see that a matrix can be seen as a linear transformation and that applying a matrix on its eigenvectors gives new vectors with same direction. Then we will see how to express quadratic equations into a matrix form. We will see that the eigendecomposition of the matrix corresponding to the quadratic equation can be used to find its minimum and maximum. As a bonus, we will also see how to visualize linear transformation in Python!
Linear algebra handwritten notes
Singular Value Decomposition. Since the beginning of this series, I emphasized the fact that you can see matrices as linear transformation in space. With the SVD, you decompose a matrix in three other matrices. Instead of doing the transformation in one movement, we decompose it in three movements. As a bonus, we will apply the SVD to image processing.
We will see the effect of SVD on an example image of Lucy the goose so keep on reading! The Moore-Penrose Pseudoinverse. We saw that not all matrices have an inverse. It is unfortunate because the inverse is used to solve system of equations. However it can be useful to find a value that is almost a solution in term of minimizing the error. This can be done with the pseudoinverse!
We will see for instance how we can find the best-fit line of a set of data points with the pseudoinverse. The Trace Operator. We will see what is the Trace of a matrix. The Determinant. This chapter is about the determinant of a matrix. This special number can tell us a lot of things about our matrix! Example: Principal Components Analysis. This is the last chapter of this series on linear algebra! We will use some knowledge that we acquired along the preceding chapters to understand this important data analysis tool!
This content is aimed at beginners but it should be easier for people with at least some experience with mathematics. I hope that you will find something interesting in that series. Engineers and scientists need to have an introduction to linear algebra in a context they understand. This book explains matrices in the detail required by engineering or science students, and it discusses linear systems of ordinary differential equations.
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Linear Algebra for Deep Learning
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